shock tube studies of thermal decomposition...

SHOCK TUBE STUDIES OF THERMAL

DECOMPOSITION REACTIONS USING

ULTRAVIOLET ABSORPTION SPECTROSCOPY

Report No. TSD-160

Matthew A. Oehlschlaeger

June 2005

© Copyright by Matthew A. Oehlschlaeger 2005

All Rights Reserved

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Abstract

Several elementary thermal decomposition reactions of importance in combustion

systems have been investigated using ultraviolet laser absorption spectroscopy and high-

temperature shock tube methods. These studies are broken into three categories: 1) the

decomposition of alkanes, 2) the incubation and decomposition of CO2, and 3) the

decomposition of the benzyl radical and toluene.

Mixtures of four alkanes dilute in argon were shock heated to determine rate

coefficients for five C-C bond fission reactions:

C2H6 → CH3 + CH3 (1)

C3H8 → CH3 + C2H5 (2)

i-C4H10 → CH3 + i-C3H7 (3)

n-C4H10 → CH3 + n-C3H7 (4)

n-C4H10 → C2H5 + C2H5 (5)

The progress of reaction was monitored by measuring methyl radical (CH3) concentration

using narrow-linewidth laser absorption at 216.62 nm. Experiments were performed over

the temperature range of 1297 to 2034 K and pressure range of 0.13 to 8.8 atm. The rate

coefficient determinations were fit using Rice-Ramsperger-Kassel-Marcus (RRKM) and

master equation calculations using a restricted (hindered) Gorin model for the transition

states. These fits provide analytic expressions for the low- and high-pressure-limit rate

coefficients and falloff parameters in the Troe pressure broadening formulation.

The thermal decomposition of CO2

CO2 + M CO + O(3P) + M (6)

was investigated behind shock waves at temperatures of 3200 to 4600 K and pressures of

0.44 to 0.98 atm. Ultraviolet laser absorption at 216.5 and 244 nm was used to monitor

the CO2 concentration with microsecond time resolution, allowing the observation of a

pronounced incubation period prior to steady CO2 dissociation for the first time,

confirming the expected bottleneck in collisional activation of this triatomic molecule.

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Master-equation calculations, with a simple model for collisional energy transfer, were

carried out to describe the measured incubation times and decomposition rate coefficient.

The master equation calculations suggest that the energy transferred per collision must

have a greater than linear dependence on energy.

The decomposition of benzyl and toluene

C6H5CH2 C7H6 + H (7)

C6H5CH3 C6H5CH2 + H (8)

C6H5CH3 C6H5 + CH3 (9)

was investigated behind shock waves at temperatures of 1398 to 1782 K and pressures

around 1.5 atm. Benzyl radicals were detected using laser absorption at 266 nm to

monitor the progress of reaction. The fast decomposition of benzyl iodide (C6H5CH2I)

dilute in argon behind shock waves provided an instantaneous benzyl (C6H5CH2) source

enabling the measurement of the rate coefficient for reaction (7) by monitoring the

pseudo-first-order decay in benzyl absorption at 266 nm. Rate coefficients for reactions

(8) and (9) were determined by monitoring benzyl absorption at 266 nm during the shock

heating of toluene dilute in argon. RRKM/master equation calculations were carried out

for the two-channel toluene decomposition using a restricted (hindered) Gorin model for

the two transition states providing extrapolation of the measured rate coefficients to the

high-pressure-limit.

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Acknowledgements

I would like to thank my advisor, Professor Ronald Hanson, for the opportunities,

support, and encouragement he has given me during my time at Stanford. His creative

ideas, critical thinking, and incredible effort have served as an excellent model for me

throughout the research process. I also would like to thank my reading committee,

Professors Tom Bowman and David Golden, for suggestions regarding the content of this

thesis.

I have been fortunate to work with many outstanding people in the Hanson

laboratory at Stanford. I am especially grateful to Dr. David Davidson for the

contributions he has made to this research and for always making the laboratory an

enjoyable place to work. I am also grateful to Dr. Jay Jeffries for the encouragement and

technical contributions he has made to this work. Both Drs. Davidson and Jeffries have

played a critical role in the development of graduate students in the Hanson laboratory

and I am grateful for their commitment, guidance, and friendship.

I am also grateful to the all of the fellow students who I have worked alongside at

Stanford. I am particularly grateful to John Herbon for teaching me many things early in

my research career, enabling me to carry out the work presented in this thesis, and Dan

Mattison for his friendship and the collaborative efforts we have made together in

research and classwork. I will always reflect fondly on my graduate days at Stanford due

mostly to the great friends with whom I was able to share the experiences.

I am particularly grateful to my family. I would like to thank my mother Deb,

father Fritz, and sister Amy for their steadfast support and encouragement. Finally, I am

most indebted to my wife and best friend, Mitra, for her support, sacrifices, and most

importantly always making me smile.

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Contents

Abstract…………………………………………………………………………….….. iii

Acknowledgements…………………………………………………………………… v

Contents……………………………………………………………………………….. vii

List of tables……………………………………………………………………….….. xi

List of figures…………………………………………………………………….……. xiii

Chapter 1: Introduction……………………………………………………………... 1

1.1 Motivation……………………………………………………………......… 1

1.2 Background………………………………………………………………… 2

1.2.1 Alkane decomposition………….……………………………………. 2

1.2.2 CO2 decomposition and incubation...……………….……………….. 3

1.2.3 Benzyl radical and toluene decomposition………….……………….. 4

1.3 Scope and organization of thesis……………………………………............6

Chapter 2: Experimental methods………………………………………………….. 11

2.1 Shock tube facility……………………………………………..…………... 11

2.2 CH3 ultraviolet laser absorption diagnostic………………………………... 13

2.2.1 Optical setup…………………………………………………………. 13

2.2.2 CH3 absorption coefficient measurements and results...…………….. 15

2.3 CO2 ultraviolet laser absorption diagnostic………………………………... 16

2.3.1 Optical setup…………………………………………………………. 17

2.3.2 CO2 ultraviolet absorption cross-section measurements…………….. 18

2.3.3 CO2 ultraviolet absorption cross-section results……………………... 19

2.3.4 CO2 ultraviolet absorption ground state energy levels………………. 20

2.4 Benzyl radical ultraviolet laser absorption diagnostic……………………... 22

2.4.1 Benzyl radical absorption cross-section measurements and results…. 22

Chapter 3: Theory and computational techniques……………………………….... 37

3.1 Unimolecular theory……………….…………………………..………...… 37

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3.2 Master equation / RRKM computational methods……………..........…….. 40

3.3 Troe formulation for falloff………………………………………………... 42

Chapter 4: Alkane decomposition…………………………………………………... 45

4.1 Determination of rate coefficients………………………………………..... 45

4.2 RRKM / master equation calculations………………………………........... 47

4.3 Analytic Arrhenius / Troe expressions for rate coefficients……………….. 48

4.4 Discussion………………………………………………………………….. 50

4.4.1 Comparison with previous experimental studies…………………….. 50

4.4.2 Comparison with previous theoretical calculations………………….. 51

4.4.3 Geometric mean rule…………………………………………………. 52

Chapter 5: CO2 incubation and decomposition……………………………………. 65

5.1 Determination of incubation times and decomposition rate coefficients…... 65

5.2 Master equation calculations…….…………………………………………. 67

5.2.1 Energy specific rate coefficient k(E)…………………………………. 68

5.2.2 Energy transfer……………………………………………………….. 69

5.3 Discussion……….…………………………………………………………. 70

5.3.1 CO2 decomposition rate coefficient………………………………….. 70

5.3.2 CO2 incubation……………………………………………………….. 71

Chapter 6: Benzyl and toluene decomposition……………………………….…….. 75

6.1 Benzyl decomposition…………………….……………………………...… 75

6.1.1 Determination of decomposition rate coefficient……………….……. 75

6.1.2 Comparison with previous studies……………………………………. 77

6.2 Toluene decomposition………………………….…………………………. 78

6.2.1 Determination of toluene decomposition rate coefficients…………… 78

6.2.2 RRKM / master equation calculations………………………………... 79

6.2.3 Comparison with previous studies……………………………………. 80

Chapter 7: Conclusions…………………………………………………………...…. 89

7.1 Summary of results………………………………………………………… 89

7.1.1 Alkane decomposition……………………………………………..… 89

7.1.2 CO2 incubation and decomposition………………………………….. 90

7.1.3 Benzyl and toluene decomposition…………………………………... 91

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7.2 Recommendations for future work………………………………………… 92

7.2.1 High-pressure ethane and propane decomposition measurements…... 92

7.2.2 Oxidation of toluene…………………………………………………. 92

7.2.3 HO2 and H2O2 reactions……………………………………………… 93

Appendix A: Rate coefficient data………………………………………………...…95

Appendix B: Uncertainty analysis…………………………………………………... 99

Appendix C: CO2 thermometry………………………………………………….….. 105

C.1 Introduction………………………………………………………………… 105

C.2 CO2 temperature diagnostic………………………………………………... 106

C.3 Experimental……………………………………………………………….. 108

C.4 Results and discussion……………………………………………………... 108

C.5 Summary…………………………………………………………………… 112

Appendix D: Azomethane synthesis………………………………………………… 119

D.1 Chemicals…………………………………………………………………... 119

D.2 Supplies…………………………………………………………………….. 119

D.3 Directions for azomethane production……………………………………... 120

D.4 Azomethane yield………………………………………………………….. 121

Appendix E: Mechanism used for benzyl and toluene decomposition…………… 123

Appendix F: Shock tube gasdynamics……………………………………………… 125

F.1 High-temperature test-time..……...………………………………………... 125

F.2 Low-temperature test-time..……...……………………………………….... 126

F.3 Reflected shock test-time and pressure traces……………………………... 126

References……………………………………………………………………………... 133

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List of tables

Table 2.1: Laser RMS intensity noise for 1 ms measurement. …………………...…… 25

Table 4.1: Parameters used in RRKM / master equation calculations for alkane

decomposition. ………………………………………………………………… 53

Table 5.1: Reaction mechanism for CO2 decomposition system. …………………….. 72

Table 5.2: Parameters used in master equation calculations for CO2 thermal

decomposition. ………………………………………………………………… 72

Table 6.1: Parameters used in RRKM / master equation calculations for toluene

decomposition. ………………………………………………………………… 82

Table A.1: Summary of experimental results for C2H6 decomposition. ………………. 95

Table A.2: Summary of experimental results for C3H8 decomposition. ………………. 96

Table A.3: Summary of experimental results for i-C4H10 decomposition. ……………. 96

Table A.4: Summary of experimental results for n-C4H10 decomposition. …………… 97

Table A.5: Summary of experimental results for CO2 decomposition and incubation. . 97

Table A.6: Summary of experimental results for benzyl radical decomposition. …….. 98

Table A.7: Summary of experimental results for toluene decomposition. ……………. 98

Table B.1: Uncertainties in k1-k9. ……………………………………………………… 100

Table B.2: Uncertainty analysis for ethane decomposition rate coefficient, k1. ………. 100

Table B.3: Uncertainty analysis for propane decomposition rate coefficient, k2. ……... 100

Table B.4: Uncertainty analysis for iso-butane decomposition rate coefficient, k3. …... 101

Table B.5: Uncertainty analysis for overall n-butane decomposition rate coefficient,

k4 + k5. …………………………………………………………………………. 101

Table B.6: Uncertainty analysis for n-butane decomposition branching ratio, k4/(k4 +

k5). ……………………………………………………………………………... 101

Table B.7: Uncertainty analysis for CO2 decomposition rate coefficient, k6. ……….… 102

Table B.8: Uncertainty analysis for benzyl decomposition rate coefficient, k7. …….… 102

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Table B.9: Uncertainty analysis for overall toluene decomposition rate coefficient, k8

+ k9. ……………………………………………………………………………. 103

Table B.10: Uncertainty analysis for overall toluene decomposition branching ratio,

k8 / (k8 + k9). ………………………………………………………..………….. 103

Table E.1: Reaction mechanism used for modeling toluene and benzyl decomposition

experiments, rate coefficients in the form k = ATb exp(-Ea / RT). …………….. 123

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List of figures

Figure 1.1: Sensitivity (Sign,i = [dτign/dki][ki/τign]) of propane ignition delay time to

various reactions, based on GRI Mech 3.0 [95]. Conditions: stoichiometric

propane in air, 1500 K, and 1.0 atm. …………...……………………………… 7

Figure 1.2: Previous experimental data for the ethane decomposition rate coefficient,

k1, normalized to 1 atm; taken from Davidson et al. (1995) [11]. …………..… 7

Figure 1.3: Previous experimental data for the propane decomposition rate

coefficient, k2. ……………………………………………………………….... 8

Figure 1.4: Previous experimental data for the iso-butane decomposition rate

coefficient, k3. ……………………………………………………………..…... 8

Figure 1.5: Previous experimental data for the benzyl radical decomposition rate

coefficient, k7. ..................................................................................................... 9

Figure 1.6: Previous experimental data for the toluene overall decomposition rate

coefficient k8 + k9. Luther et al. [94] performed a laser excitation study to

determine the rate coefficient at infinite pressure. ………………..……..…….. 9

Figure 1.7: Previous experimental data for the toluene decomposition branching ratio

k8/(k8 + k9). Luther et al. [94] performed a laser excitation study to determine

the branching ratio at infinite pressure. …………………..…………………… 10

Figure 2.1: Absorption spectra of CH3 B2A1’ X2A2

’’ electronic system at 1565 K,

Oehlschlaeger et al. [118]. …………………………………………………….. 26

Figure 2.2: Schematic of 216.62 nm experimental setup: M, mirror; BS, beam splitter;

L, lens; D, detector; PT, piezoelectric pressure transducer; and Kistler,

shielded Kistler piezoelectric pressure transducer. …………………………..... 26

Figure 2.3: Laser intensity noise measured with Thor Labs PDA55 detector and

Hamamatsu S1722-02 silicon photodiode. Top graph, Ar+ pump output (all-

lines UV), RMS noise is ±0.098%; middle graph, dye laser output (433.24

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nm), RMS noise is ±0.15%; bottom graph, doubling cavity output (216.62

nm), RMS noise is ±0.78%. ………………………………………………….... 27

Figure 2.4: Absorbance signal at 216.62 nm using two-beam common-mode rejection

with Thor Labs PDA55 detectors and Hamamatsu S1722-02 silicon

photodiodes, RMS noise is ±0.09%. ……………………...………………...…. 28

Figure 2.5: Modified Thorlabs PDA55 (Hamamatsu S1722-02 silicon photodiode)

detector spatial sensitivity at 266 nm after a focused 1 mW 266 nm beam was

incident on the photodiode surface. …………………………………………… 28

Figure 2.6: Example azomethane decomposition data and modeling. Reflected shock

conditions: 1561 K, 1.622 atm, 200 ppm azomethane/Ar; laser wavelength =

216.62 nm, kλ,CH3 = 34.3 cm-1atm-1. ……………………….………..…………. 29

Figure 2.7: Temperature dependent CH3 absorption coefficient at 216.62 nm. Solid

squares, azomethane data; solid circles, methyl iodide data; solid triangles,

ethane data; solid line, fit from this study; x, Davidson et al. [101]; open

circles, Tsuboi [104]; dashed line, Möller et al. [103]; dotted line, Glänzer et

al. [102]; dash-dot line Du et al. [106] (215.94 nm); dash-dot-dot line, Hwang

et al. [105] (213.9 nm). ………………………………………………………... 29

Figure 2.8: Schematic of 244 and 266 nm experimental setup: L, lens; BBO, beta

barium borate frequency doubling crystal; PB, Pellin-Broca prism; M, mirror;

BS, beam splitter; D, detector; PT, piezoelectric pressure transducer; and

Kistler, shielded Kister piezoelectric pressure transducer. .…………………… 30

Figure 2.9: Laser intensity noise measured with Thor Labs PDA55 detector and

Hamamatsu S1722-02 silicon photodiode. Top graph, 244 nm output (doubled

488 nm Ar+ line), RMS noise is ±0.13%; bottom graph, 266 nm output

(doubled 532 nm Nd:YVO4), RMS noise is ±0.06%. …………………………. 31

Figure 2.10: Example CO2 absorbance for experiments at 216.5, 244, 266, and 306

nm. .……………………………………………………………………………. 32

Figure 2.11: Example CO2 absorbance at 216.5 nm for experiment with thermal

decomposition. Initial mixture: 2% CO2 / Ar. Vibrationally equilibrated

incident shock conditions: 1812 K and 0.161 atm. Initial (prior to

decomposition) vibrationally equilibrated reflected shock conditions: 3838 K,

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0.820 atm. Note the vibrational relaxation after incident shock-heating and the

incubation period prior to decomposition after reflected shock-heating. ..…..... 32

Figure 2.12: Temperature-dependence of the UV CO2 absorption cross-section. Solid

symbols, experimental results: squares, 216.5 nm; circles, 244 nm; triangles,

266 nm; diamonds, 306 nm. Solid lines, semi-empirical fit to current data;

dashed lines, Schulz et al. [112]. ………….…………………………………… 33

Figure 2.13: Wavelength-dependence of the UV CO2 absorption cross-section. Solid

symbols, semi-empirical fit to current data: squares, 3050 K; circles, 2610 K,

diamonds, 2010 K; stars, 1630 K. Solid lines, Schulz et al. [112]. …………… 33

Figure 2.14: Product of cross-section and vibrational partition function (σCO2Qvib)

versus inverse temperature (top graph) and ground state energy versus

wavelength (bottom graph). Top graph: solid squares, 216.5 nm data; solid

circles, 244 nm data; solid triangles, 266 nm data; solid diamonds, 306 nm

data; solid lines, least-squares fit of form σCO2(λ,T)Qvib = σCO2,eff(λ)exp(-ε1/T).

Bottom graph: squares, current results; line, linear fit to current results; open

circles, Eremin et al. [114]; x, Zabelinskii et al. [115]. ……………………….. 34

Figure 2.15: Benzyl radical spectrum at 1600 K, Müller-Markgraf and Troe [82]. …... 35

Figure 2.16: Example 266 nm absorbance during benzyl iodide decomposition.

Reflected shock conditions: 1615 K, 1.556 atm, 50 ppm benzyl iodide/Ar. ….. 35

Figure 2.17: Benzyl radical absorption cross-section at 266 nm. Solid squares, this

study; open circle, Ikeda et al. [116]; open triangle, Müller-Markgraf and

Troe [82]. .……………………………………………………………………... 36

Figure 2.18: Benzyl fragment absorption cross-section at 266 nm. Solid squares, this

study; open triangle, Müller-Markgraf and Troe [82]. ………………………... 36

Figure 3.1: Reactant population energy distribution under high-pressure-limit

conditions (solid) and under falloff conditions (dashed). Under falloff

conditions the unimolecular dissociation step occurs at a rate fast enough such

that the equilibrium Boltzmann distribution cannot be maintained by

collisions and the population is depleted above the reaction threshold. ...…….. 43

Figure 4.1: Example ethane data, modeling, and sensitivity. Reflected shock

conditions: 1589 K, 1.664 atm, 202 ppm C2H6. Top graph: solid line, fit to

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data using GRI-Mech 3.0 [95] and adjusting k1; dashed lines, variation of k1

±50%. The spike at 0 µs is caused by deflection of the diagnostic beam by the

reflected shock wave. S = (dXCH3 / dki) (ki / XCH3,local), where ki is the rate

constant for reaction i and XCH3,local is the local CH3 mole fraction. ………...… 55

Figure 4.2: Example propane data, modeling, and sensitivity. Reflected shock

conditions: 1433 K, 4.535 atm, 201 ppm C3H8. Top graph: solid line, fit to

data using GRI-Mech 3.0 [95] and adjusting k2; dashed lines, variation of k2

±50%; dotted lines, variation of rate coefficient for H + C3H8 C3H7 + H2 by

±50%. S = (dXCH3 / dki) (ki / XCH3,local), where ki is the rate constant for

reaction i and XCH3,local is the local CH3 mole fraction. ……………..…………. 55

Figure 4.3: Example iso-butane data, modeling (top), and sensitivity (bottom).

Incident shock conditions: 1401 K, 0.238 atm, 400 ppm i-C4H10/Ar. Top

graph: solid line, fit to data using Wang et al. [133] mechanism and adjusting

k3; dashed lines, variation of k3 ±50%; dotted lines, variation of rate

coefficient for i-C4H10 + H i-C4H9 + H2 by ±50%. S = (dXCH3 / dki) (ki /

XCH3,local), where ki is the rate constant for reaction i and XCH3,local is the local

CH3 mole fraction. ……………………………………………………….……. 56

Figure 4.4: Example n-butane data, modeling (top), and sensitivity (bottom).

Reflected shock conditions: 1479 K, 1.673 atm, 200 ppm n-C4H10/Ar. Top

graph: solid line, fit to data using Marinov et al. [134] mechanism and

adjusting overall decomposition rate, k4+k5, and branching ratio, k4/(k4+k5);

dashed lines, variation of k4+k5 ±50%; dotted lines, variation of k4/(k4+k5)

±25%. S = (dXCH3 / dki) (ki / XCH3,local), where ki is the rate constant for

reaction i and XCH3,local is the local CH3 mole fraction. …….………………….. 56

Figure 4.5: Experimental data and least-squares fits for C2H6 CH3 + CH3. Filled

squares, 1.6 atm data, k1 = 1.08x1012 exp(-31200K/T); filled triangles, 4.2 atm

data, k1 = 4.01x1012 exp(-32400K/T); filled stars, 7.5 atm data, k1 = 1.55x1013

exp(-34200K/T); open squares, 0.13-0.22 atm data. …………………………... 57

Figure 4.6: Experimental data and least-squares fits for C3H8 CH3 + C2H5. Filled

squares, 1.7 atm data, k2 = 1.97x1013 exp(-32400K/T); filled triangles, 4.3 atm

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data, k2 = 5.10x1013 exp(-33400K/T); filled stars, 7.9 atm data, k2 = 4.97x1013

exp(-33100K/T); open squares, 0.19-0.23 atm data. …………….…………….. 57

Figure 4.7: Experimental data and least-squares fits for i-C4H10 CH3 + i-C3H7.

Filled squares, 1.7 atm data, k3 = 4.98x1014 exp(-34800K/T) s-1; filled

triangles, 4.4 atm data, k3 = 1.71x1015 exp(-36300K/T) s-1; filled stars, 8.4 atm

data, k3 = 3.59x1015 exp(-37000K/T) s-1; open squares, 0.21-0.25 atm data. …. 58

Figure 4.8: Experimental data and least-squares fits for n-C4H10 CH3 + C3H7.




Figure 4.9: Experimental data and least-squares fits for n-C4H10 C2H5 + C2H5.




Figure 4.10: Hindrance parameter temperature dependence. Filled squares, RRKM fit

to k1 data of current study; filled circles, RRKM fit to k2 data of current study;

open squares, RRKM fit to k1 data of Slagle et al. [23]; open circles, RRKM

fit to k2 data of Knyazev and Slagle [32]. ……………………...……………… 59

Figure 4.11: Falloff plot for C2H6 CH3 + CH3. Solid symbols, current study; open

symbols, Davidson et al. [11]; open symbols with x, Du et al. [13]; half filled

symbols, Hwang et al. [12]; solid lines, RRKM/master equation results with α

= 0.8 x (T [K]) cm-1; dashed lines, Baulch et al. [201] IUPAC

recommendation . …………………………………….………………………... 60

Figure 4.12: Falloff plot for C3H8 CH3 + C2H5. Solid symbols, current study; open

symbols, Al-Alami and Kiefer [27]; half filled symbols, Simmie et al. [29];

open symbols with line, Hidaka et al. [30]; open symbols with x, Chiang and

Skinner [26]; solid lines, RRKM/master equation results with α = 1.2 x (T

[K]) cm-1; dashed lines, Baulch et al. [201] IUPAC recommendation. …….…. 60

Figure 4.13: Falloff plot for i-C4H10 CH3 + i-C3H7. Solid symbols, current study;

open symbols, Hidaka et al. [35]; half filled symbols, Koike and Morinaga

[36]; solid lines, RRKM/master equation results with α = 0.8 x (T [K]) cm-1. ... 61

xvii

Figure 4.14: Falloff plot for n-C4H10 CH3 + n-C3H7. Solid symbols, current study;

open symbols, Koike and Morinaga [37]; solid lines, RRKM/master equation

results with α = 1.2 x (T [K]) cm-1. ……………………………………………. 61

Figure 4.15: Falloff plot for n-C4H10 C2H5 + C2H5. Solid symbols, current study;

solid lines, RRKM/master equation results with α = 1.2 x (T [K]) cm-1. …...… 62

Figure 4.16: High pressure rate coefficient for C2H6 CH3 + CH3 (700-1924K).

Symbols, current RRKM results (listed on figure); solid line, fit to current

RRKM results; dashed line, Klippenstein and Harding [22]; dotted line,

Hessler and Ogren [20]; dot-dashed line, Wagner and Wardlaw [15]; gray

line, Baulch et al. [201] IUPAC recommendation. …………………………..... 62

Figure 4.17: High pressure rate coefficient for C3H8 CH3 + C2H5 (600-1653 K).

Symbols, current RRKM results (listed on figure); solid line, fit to current

RRKM results; dashed line, Mousavipour and Homayoon [33]; dotted line,

Knyazev and Slagle [32]; dot-dashed line, Tsang [17]; dot-dot-dashed,

Dean [14] ; gray line, Baulch et al. [201] IUPAC recommendation. …..…...…. 63

Figure 4.18: High-pressure rate coefficient for i-C4H10 CH3 + i-C3H7. Symbols,

current RRKM results (listed on figure); solid line, fit to current RRKM

results; dashed line, Tsang [17]; dotted line, Warnatz [153]. …………………. 63

Figure 4.19: High-pressure rate coefficient for n-C4H10 CH3 + n-C3H7. Filled

squares, current RRKM results (listed on figure); solid line, fit to current

RRKM results; dashed line, Dean [14]; dotted line, Warnatz [153]. ………….. 64

Figure 4.20: High-pressure rate coefficient for n-C4H10 C2H5 + C2H5. Filled

circles, current RRKM results (listed on figure); solid line, fit to current

RRKM results; dashed-dot line, Dean [14]; dashed-dot-dot line, Warnatz

[153]; dashed line, result using the geometric mean rule. ………………….…. 64

Figure 5.1: Example experiment and fit (dark line): CO2 absorbance (ln(I0/I)) at 216.5

nm. Initial mixture 2% CO2/Ar. Vibrationally equilibrated shock conditions:

1812 K and 0.161 atm. Initial (prior to decomposition) vibrationally

equilibrated reflected shock conditions: 3838 K, 0.820 atm. ………………..... 72

Figure 5.2: Second order rate coefficient for CO2 + Ar CO + O + Ar: filled squares

with error bars, current experimental results; heavy solid line, fit to current

xviii

data; dashed line, Burmeister and Roth [41]; solid line, Fujii et al. [42]; dash-

dot line, Ebrahim and Sandeman [43]; short dash line, Hardy et al. [44]; dash-

dot-dot line, Kiefer [46]; smaller dotted line, Dean [47]. ……………...……… 73

Figure 5.3: Number of Lennard-Jones collisions in CO2-Ar incubation period

(ZL-J[M]∆tinc): experiment, open circles with error bars; master equation, filled

squares. …………………………………………………………………………73

Figure 5.4: Example master equation result for experiment denoted with an asterisk

(*) in Table A.5. Initial conditions: Ttrans = Trot = 4171 K, Tvib =1943 K, and P

= 0.735 atm. Open circles, master equation calculation; solid line, fit to

calculation for determination of incubation time and rate coefficient. ………... 74

Figure 6.1: Example 266 nm absorbance during benzyl iodide decomposition.

Reflected shock conditions: 1615 K, 1.556 atm, 50 ppm benzyl iodide/Ar.

Solid line, best fit to absorbance by adjusting k7; dashed line, contribution to

absorbance from benzyl; dotted line, contribution to absorbance from benzyl

fragments; dot-dashed line, contribution to absorbance from toluene. ………... 83

Figure 6.2: Local sensitivity for benzyl concentration for conditions of experiment

given in Figure 6.1. S = (dXbenzyl / dki) (ki / Xbenzyl,local), where ki is the rate

constant for reaction i and Xbenzyl,local is the local benzyl (C6H5CH2) mole

fraction. …………………………………………………………………….….. 83

Figure 6.3: Rate coefficient for benzyl decomposition, reaction (7): filled squares

with error bars, current experimental results (~1.5 atm); heavy solid line, fit to

current data; gray line, Hippler et al. [81] (~0.5 atm); dashed line, Braun-

Unkhoff et al. [87] (~2 atm) and Baulch et al. [201]; dotted line, Jones et al.

[80] (~10-12 atm); dash-dot line, Rao and Skinner [86] (~0.6 atm). ………….. 84

Figure 6.4: Example 266 nm absorbance during toluene decomposition. Reflected

shock conditions: 1593 K, 1.461 atm, 200 ppm toluene / Ar. Solid line, fit to

data by adjusting overall decomposition rate, k8+k9, and branching ratio,

k8/(k8+k9); dotted lines, variation of k8+k9 ±50%; dashed lines, variation of

k8/(k8+k9) ±30%. ……………………………………………………………….. 84

xix

Figure 6.5: Contribution to 266 nm absorbance from benzyl (dashed line), benzyl

fragment (dashed-dotted line), and toluene (dotted line) for example

experiment given in Figure 6.4. …………………………….…………………. 85

Figure 6.6: Local sensitivity for benzyl concentration for conditions of experiment

given in Figure 6.4. S = (dXbenzyl / dki) (ki / Xbenzyl,local), where ki is the rate

constant for reaction i and Xbenzyl,local is the local benzyl (C6H5CH2) mole

fraction. …………………………………………………………………….….. 85

Figure 6.7: Rate coefficient for C6H5CH3 C6H5CH2 + H, reaction (8): filled

squares with error bars, current experimental results (~1.5 atm); solid gray

line, fit to current data; solid black line, RRKM / master equation result for

k8,∞; dashed line, RRKM / master equation result for k8(1.5 atm); dotted line,

Baulch et al. [201] IUPAC recommendation for k8,∞. …………………………. 86

Figure 6.8: Rate coefficient for C6H5CH3 C6H5 + CH3, reaction (9): filled circles

with error bars, current experimental results (~1.5 atm); solid gray line, fit to

current data; solid black line, RRKM / master equation result for k9,∞; dashed

line, RRKM / master equation result for k9(1.5 atm); dotted line, Baulch et al.

[201] IUPAC recommendation for k9,∞. ……………………………….………. 86

Figure 6.9: Comparison with previous data for the toluene overall decomposition rate

coefficient k8 + k9. Luther et al. [94] performed a laser excitation study to

determine the rate coefficient at infinite pressure. ..………………………….... 87

Figure 6.10: Comparison with previous data for the toluene decomposition branching

ratio k8 / (k8 + k9). Luther et al. [94] performed a laser excitation study to

determine the branching ratio at infinite pressure. …………….......................... 87

Figure C.1: Ratio of 266 to 244 nm absorption cross-section; also referred to as

absorbance ratio, R. ……………………………………………………………. 113

Figure C.2: Uncertainty in measured temperature for given temperature and product

of partial CO2 pressure and path length (PCO2L). The contours represent lines

of constant uncertainty in the temperature measurement assuming an

uncertainty in measured absorbance of ±0.1% (-ln(I / I0) = ±0.001). ……….… 113

xx

Figure C.3: Example non-reacting experiment: measured absorbance at 266 and 244

nm and measured pressure. Initial reflected shock conditions from ideal shock

relations: 10% CO2/Ar, 2138 K and 1.226 atm. .……………………………… 114

Figure C.4: Measured temperature (dark line) vs isentropic temperature (light line) for

experiment in Figure C.3. Measured temperature determined from absorbance

ratio (absorbance traces given in Figure C.3). Isentropic temperature

determined from pressure measurement (given in Figure C.3). ………………. 114

Figure C.5: Comparison of measured temperature to isentropic temperature from the

results given in Figure C.4. ……………………………………………………. 115

Figure C.6: Comparison of measured post-shock temperature to temperature

calculated with the ideal shock relations. Post-shock vibrational equilibrium

was assumed. ……………………………………………………………….. 115

Figure C.7: Molecular oxygen absorption cross-section at 244.061 and 266.075 nm (1

atm). ….………………………………………………………………………... 116

Figure C.8: Molecular oxygen absorption spectrum around 244 nm for 2000K and 1

atm, calculated using LIFSIM calculator [187]. Current doubled Ar+ line is at

244.061 nm, a source at 243.99 nm would reduce the interference due to O2

by a factor of seven. …………………………………………………………… 116

Figure C.9: Example absorbance for energetic ignition experiment. Initial reflected

shock conditions: 1% CH4/ 2% O2/ 10% CO2/ Ar, 2085 K and 1.297 atm. 266

nm and 244 nm measured absorbance, dark lines; corrected (O2 interference)

244 nm absorbance, light line. ……………………………………...…………. 117

Figure C.10: Measured temperature (solid thick lines) versus constant volume

calculation (solid thin lines). Top graph (initial reflected shock conditions):

1% CH4/ 2% O2/ 10% CO2/ Ar, 2085 K, and 1.297 atm. Middle graph: 2%

CH4/ 4% O2/ 10% CO2/ Ar, 1890 K, 1.358 atm. Bottom graph: 4% CH4/ 8%

O2/ 10% CO2/ Ar, 1851 K, 1.277 atm. Post-shock vibrational equilibrium was

assumed. Error bars represent a 1-σ confidence interval. ……………………... 118

Figure F.1: Schematic of x-t diagram for high-temperature (strong shock) case where

a2 > a3. ……………………………………………………………………….… 128

xxi

Figure F.2: Schematic of x-t diagram for high-temperature (strong shock) case where

a3 > a3. …………………………………………………………………………. 128

Figure F.3: Schematic x-t diagram for high-temperature (strong shock) case where a2

= a3, tailored condition. ………………………………………………………... 129

Figure F.4: Schematic x-t diagram for low-temperature (weak shock) case where

reflected shock interaction with rarefaction wave limits test-time. …………… 129

Figure F.5: Example pressure traces measured with KISTLER piezoelectric

transducer. Top graph: example high-temperature (strong shock) case where

a2 > a3 (Figure F.1 x-t diagram). Middle graph: example high-temperature

(strong shock) case where a3 > a2 (Figure F.2 x-t diagram). Bottom graph:

example low-temperature case (Figure F.4 x-t diagram). ..……………………. 130

Figure F.6: Measured test-time as a function of reflected shock temperature for

helium driven argon shocks. ………................................................................... 131

xxii

Chapter 1: Introduction

1.1 Motivation The design and optimization of combustion systems relies heavily on accurate

predictive modeling. Models for combustion systems typically involve mathematical

solution of a set of partial and ordinary differential equations describing heat transport,

fluid dynamics, and chemistry. Solution of these equations provides information

regarding performance such as efficiency and pollutant emissions. The importance of

chemistry in modeling combustion systems varies significantly depending on the type of

combustion system. For instance, in a highly turbulent combusting flow the fluid

dynamics may play a more important role than the chemistry and a kinetic model might

be able to predict important observables with a reduced chemistry scheme or even a one-

step global reaction rate. However, in many combustion systems the rates of individual

elementary chemical reactions are significant in governing overall performance. For

instance, the ignition timing, and therefore stability, of a homogenous charge

compression ignition (HCCI) engine is governed by chemical kinetics, not transport

phenomena. Optimization of advanced combustion systems, such as HCCI, can only be

realized when the chemical reactions that govern their performance are well understood.

One class of reactions important in describing the overall ignition characteristics

of a fuel/oxidizer mixture are thermal decomposition reactions (also called dissociation

reactions). These reactions are often responsible for the initial fragmentation of a fuel

undergoing oxidation and therefore the overall rate of ignition, or ignition delay time, of a

given fuel/oxidizer mixture is often sensitive to the rates of decomposition reactions. In

addition, thermal decomposition reaction rate coefficients have the interesting

characteristic of exhibiting pressure-dependence, providing another complication in the

modeling and measurement of their rates.

The subject of the studies presented here is the measurement of reaction rate

coefficients for several important thermal decomposition reactions. These studies are

1

broken up into three main topics: 1) the decomposition of alkanes, 2) the decomposition

and incubation of CO2, and 3) the decomposition of benzyl radicals and toluene. In all of

these studies shock-heating was used to initiate thermal decomposition and ultraviolet

laser absorption diagnostics were used to monitor either product or reactant species.

1.2 Background 1.2.1 Alkane decomposition The thermal decomposition of alkanes plays an important role in alkane oxidation,

particularly at high-temperatures (>1000 K) where C-C fission reactions often are the

dominant initiation steps. In addition the reverse reactions, alkyl radical recombination,

play an important role as termination steps. The importance of these reactions is

quantified by the large sensitivity that global combustion phenomena, such as ignition

delay and flame speed, show for the reaction rate coefficients of these

decomposition/recombination reactions [1-2]. The following alkane decomposition

reactions are the focus of this study:

C2H6 → CH3 + CH3 (1)

C3H8 → CH3 + C2H5 (2)

i-C4H10 → CH3 + i-C3H7 (3)

n-C4H10 → CH3 + n-C3H7 (4)

n-C4H10 → C2H5 + C2H5 (5)

As an example, the sensitivity of propane ignition delay time on various reactions is

shown in Figure 1.1, where the ignition delay time is defined as the time to the peak rate

of change in the OH radical concentration. At 1500 K and 1.0 atm the ignition delay

shows the greatest sensitivity to propane decomposition, reaction (2). The ignition delays

of other alkanes show similar sensitivity to decomposition reactions motivating this study

[1-2].

Reaction (1) has been extensively studied in both the dissociation and

recombination directions across a broad range of temperatures. Previous high-

temperature shock tube measurements have utilized different techniques: gas

chromatography in a single-pulse shock tube [3], H-atom atomic resonance absorption

2

spectroscopy (ARAS) [4-5], infrared absorption of ethane at 3.39 µm [6], xenon lamp

absorption of CH3 [7], laser-schlieren [8-9], and laser absorption of CH3 [10-13].

Reaction (1) has also been the subject of many theoretical studies of varying levels of

complexity [14-22,92]. At low-temperature (<1000 K) the theoretical studies agree with

one another and with the experimental data quite well, due in a large part to the excellent

low-scatter recombination data of Slagle et al. [23] (296-906 K), but at higher

temperatures the theoretical high-pressure rate coefficients show some disagreement with

one another and experimental data. High-temperature data with very low scatter are thus

needed, in order to resolve these disagreements. A comparison of the results of the

previous experimental studies at high-temperatures for the rate coefficient of reaction (1),

k1, is shown in Figure 1.2.

Reaction (2) has been studied previously to a lesser degree than reaction (1), with

previous high-temperature studies including single-pulse shock tube studies [24-25] and

several studies utilizing time-resolved techniques: H-atom ARAS [26], laser-schlieren

[27], and 3.39 µm absorption of propane [28-30]; additionally, a recent recombination

study of reaction (-2) has been performed by Knyazev and Slagle [31] over the

temperature range 301 to 800 K. Reaction (2) has also been examined using various

levels of unimolecular theory [14,17,31-33]. The results of the previous experimental

studies on propane decomposition are compared in Figure 1.3.

Reaction (3) has been studied previously at high-temperature using single-pulse

shock tube techniques [34], 3.39 µm absorption of iso-butane [35], and UV lamp

absorption of CH3 [36]. The results of these previous experimental studies for k3 are

shown in Figure 1.4. Reactions (4) and (5) have not been studied in great detail at high-

temperatures, although there is one shock tube study of note in which UV lamp

absorption of CH3 was employed to determine k4 [37]. The reaction (5) rate coefficient

has not been previously determined at high-temperatures but, there have been several

low-temperature studies performed primarily in flow tubes [38-40].

1.2.2 CO2 decomposition and incubation The spin forbidden thermal decomposition of carbon dioxide

CO2 + M → CO + O(3P) + M (6)

3

behind shock waves has been the subject of many experimental studies [41-56], dating

back to the 1960’s, due to interest in modeling hypersonic flows and very high-

temperature combustion phenomena such as detonation waves. While several of the

experimental studies agree on the rate coefficient for CO2 decomposition, none have

observed an incubation period prior to decomposition. Despite the lack of experimental

findings for CO2 incubation, two analytical master equation calculations have predicted

such a phenomenon [57-58].

When a molecule is rapidly heated by the passing of a shock wave, energy in the

translational and rotational modes equilibrates quite rapidly (µs), but the vibrational

modes equilibrate at a somewhat slower rate that can often be observed [59-74]. At

temperatures high enough for the molecule to dissociate, this finite rate of energy transfer

leads to the observation of a delay in the onset of steady dissociation, and this delay is

defined as the incubation time [59]. Although observed for several diatomic species (e.g.,

O2 [60-62], N2 [63], CO [64]), incubation times have only been observed for a few

polyatomic species. The common characteristic required to observe incubation in a

polyatomic molecule has been a relatively high lowest vibrational frequency, which

causes a bottleneck in collisional activation. The first observation of incubation in a

polyatomic was reported by Dove et al. [65] in N2O decomposition behind shock waves.

These experimental findings were later analyzed using master equation techniques by

Forst and Penner [57], Yau and Pritchard [58], and Dove and Troe [66]. Kiefer and co-

workers also have observed incubation times in several hydrocarbons using the laser-

schlieren technique [67-70]. Barker and co-workers analyzed the Kiefer results for

cyclohexene and norbornene using numerical master equation techniques [71-72].

Recently Hippler and co-workers have observed incubation times in CH3 [73] and toluene

decomposition [74] and performed numerical master equation analysis for these

reactions.

1.2.3 Benzyl radical and toluene decomposition Aromatic fuels have received considerable attention in the past few years due to

interest in developing detailed mechanisms to describe their oxidation [75-79]. Although

detailed mechanisms are in development, there is a lack of high-quality experimental rate

4

coefficient data for the reactions of importance in describing aromatic oxidation. One

class of reactions that is poorly understood and is of importance in these mechanisms is

the thermal decomposition of aromatics.

For instance, the decomposition of benzyl radicals

C6H5CH2 → C7H6 + H (7)

plays a major role in the oxidation of aromatic hydrocarbons, particularly toluene, as a

source of H-atoms. As is the case of most reactions involving aromatics, benzyl

decomposition is not well understood and previous rate coefficient measurements show

large scatter and disagreement. Previous high-temperature measurements have been made

behind shock waves using ultraviolet absorption [80-85] and H-atom ARAS [86-87]. A

comparison of the findings of these previous studies is given in Figure 1.5.

The decomposition of toluene is one of the most important initiation steps in the

oxidation of toluene at moderate- to high-temperatures (>1200 K). Despite its importance

the decomposition of toluene is not well understood due to complications in its

measurement. The primary complication is that toluene decomposition takes place via

two channels

C6H5CH3 → C6H5CH2 + H (8)

C6H5CH3 → C6H5 + CH3 (9)

making isolated measurement of the two rate coefficients, k8 and k9, difficult.

Additionally, fast secondary and recombination reactions occur in parallel with the

decomposition, further complicating kinetic isolation of reactions (8) and (9). Previous

measurements of toluene decomposition have been made at high-temperatures behind

shock waves using time-of-flight mass spectrometry [89], laser-schlieren [89], ultraviolet

absorption [81,83,84,85,91], and H-atom ARAS [74,88,90]. Additionally, laser-excitation

techniques used to measure energy specific rate coefficients for toluene decomposition

have been performed [93-94]. A comparison of the results of these studies for the overall

decomposition rate, k8 + k9, is shown in Figure 1.6 and the branching ratio, k8/(k8 + k9), in

Figure 1.7.

5

1.3 Scope and organization of thesis The objectives of this work were to make low-uncertainty measurements of the

rate coefficients for reactions (1)-(9). To achieve this goal ultraviolet laser absorption

diagnostics were developed and spectroscopic measurements carried out for the detection

of methyl radicals (CH3), carbon dioxide (CO2), and benzyl radicals (C6H5CH2). These

laser diagnostics allowed measurement of rate coefficients for reactions (1)-(9) at high-

temperatures behind shock waves. The resulting rate coefficient data (and in the case of

CO2, incubation times) were fit using computational techniques based on Rice-

Ramsperger-Kassel-Marcus (RRKM) theory and the master equation for internal energy.

In addition to the chemical kinetic studies outlined above, a thermometry technique was

developed, using ultraviolet CO2 absorption, to characterize the temperature time-history

of shock-heated mixtures.

In Chapter 2 a description of the shock tube and laser diagnostics used in these

studies is given. Additionally, Chapter 2 contains a description of the spectroscopic

measurements made to enable determination of the reaction rate coefficients. Chapter 3

describes the basis for the RRKM/master equation computational techniques used to fit

the experimentally determined rate coefficients. Chapters 4-6 contain the major findings

of these studies, the determination of rate coefficients for reactions (1)-(9). Chapter 4

contains the findings for alkane decomposition, Chapter 5 the findings for CO2

decomposition and incubation, and Chapter 6 the findings for benzyl radical and toluene

decomposition. Conclusions and recommendations for future work are made in Chapter

7.

Appendix A contains tabulation of all of the rate coefficient and incubation time

determinations presented in Chapters 4-6. Appendix B contains the uncertainty analysis

for the rate coefficient determinations for reactions (1)-(9). A thermometry technique

using ultraviolet CO2 absorption and its application to non-reacting and reacting shock-

heated mixtures is described in Appendix C. Instructions for the synthesis of azomethane

are given in Appendix D. The reaction mechanism used for the modeling of the benzyl

and toluene decomposition experiments is given in Appendix E. Finally, Appendix F

provides x-t diagrams and a discussion of the wave dynamics and test time for the shock

tube used in the studies presented in this thesis.

6

C3H8=CH3+C2H5

H+O2=O+OH

H+C3H8=C3H7+H2

HO2+CH3=OH+CH3O

C2H5+O2=HO2+C2H4

OH+C3H8=C3H7+H2O

C2H6=2CH3

CH3O+O2=HO2+CH2O

H+CH2O=CH3O

H+C2H4=C2H5

OH+C2H4=C2H3+H2O

-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

Ignition Time Sensitivity, Sign

Figure 1.1: Sensitivity (Sign,i = [dτign/dki][ki/τign]) of propane ignition delay time to various reactions, based on GRI Mech 3.0 [95]. Conditions: stoichiometric propane in air, 1500 K, and 1.0 atm.

0.50 0.55 0.60 0.65 0.70101

102

103

104

105

106

Burcat et al. [3] Tsuboi [7] Olson et al. [9] Roth and Just [5] Chiang and Skinner [4] Kiefer and Budach [8] Hidaka et al. [6] Davidson et al. [11]

k 1 [1/

s]

1000/T [1/K]

Figure 1.2: Previous experimental data for the ethane decomposition rate coefficient, k1, normalized to 1 atm; taken from Davidson et al. (1995) [11].

7

0.50 0.55 0.60 0.65 0.70 0.75 0.80101

102

103

104

105

106

Al-Alami and Kiefer [27], 0.13-0.68 atm Hidaka et al. [30], 1.5-2.6 atm Chiang and Skinner [26], 2.0-3.0 atm Simmie et al. [29], 1.6-2.0 atm

k 2 [1/s

]

1000/T [1/K]

Figure 1.3: Previous experimental data for the propane decomposition rate coefficient, k2.

0.50 0.55 0.60 0.65 0.70 0.75 0.80101

102

103

104

105

106

k 3 [1/s

]

1000/T [1/K]

Hidaka et al. [35], 1.3-2.9 atm Koike and Morinaga [36], 0.5 atm Bradley [34], 0.7-0.9 atm

Figure 1.4: Previous experimental data for the iso-butane decomposition rate coefficient, k3.

8

0.50 0.55 0.60 0.65 0.70 0.75 0.80101

102

103

104

105

106

k 7 [1/s

]

1000/T [1/K]

Hippler et al. [81], 0.4-0.9 atm Braun-Unkhoff et al. [87], 2 atm Rao and Skinner [86], 0.6 atm Jones et al. [80], 10-12 atm

Figure 1.5: Previous experimental data for the benzyl radical decomposition rate coefficient, k7.

0.50 0.55 0.60 0.65 0.70 0.75 0.80101

102

103

104

105

106

k 8 + k

9 [1/s

]

1000/T [1/K]

Pamidimukkala et al. [89], 0.5 atm Rao and Skinner [90], 0.4 atm Luther et al. [94], infinite p Eng et al. [74], 2 atm Braun-Unkhoff et al. [88], 2 atm

Figure 1.6: Previous experimental data for the toluene overall decomposition rate coefficient k8 + k9. Luther et al. [94] performed a laser excitation study to the determine rate coefficient at infinite pressure.

9

1200 1400 1600 1800 2000

0.0

0.2

0.4

0.6

0.8

1.0

k 8 / (k

8 + k

9)

Temperature [K]

Pamidimukkala [89], 0.5 atm Rao and Skinner [90], 0.4 atm Braun-Unkhoff et al. [88], 2 atm Luther et al. [94], infinite p Eng et al. [74], 2 atm

Figure 1.7: Previous experimental data for the toluene decomposition branching ratio k8/(k8 + k9). Luther et al. [94] performed a laser excitation study to determine the branching ratio at infinite pressure.

10

Chapter 2: Experimental methods

This chapter describes the shock tube facility and the laser diagnostics used for

the measurement of reaction rate coefficients, k1-k9. The three sections on laser

diagnostics also describe the measurement of absorption cross-sections for the three

species of interest (CH3, CO2, and benzyl).

2.1 Shock tube facility The experiments reported in this thesis were performed behind both incident and

reflected shock waves in a helium pressure-driven stainless-steel shock tube. The driven

section is 8.54 m long and the driver section 3.35 m long; both sections have an inner

diameter of 14.13 cm. The shock tube driven and driver sections are separated by a

polycarbonate diaphragm 0.005 to 0.08” in thickness. The diaphragm is ruptured as the

driver section is filled with helium by a set of cutter blades configured in a cross

geometry. The bursting pressure of the diaphragm is controlled by the thickness of the

diaphragm and the position of the cutter blades.

Prior to each experiment the driven section was evacuated with a vacuum system

consisting of a zeolite-trapped mechanical pump and a Varian V-250 turbomolecular

pump, providing ultimate pressures of 10-7 torr with a typical leak rate of 10-6 torr per

minute (when evacuated overnight). Shock velocities were measured with five

piezoelectric pressure transducers (PCB 113A26 transducer, PCB 483B08 amplifier)

spaced axially over the last 1.5 m of the tube. The signals from the five transducers were

sent to four Philips PM6666 counter timers with resolution of 0.1 µs to determine the

shock passage time interval. The shock velocities measured from the shock passage time

intervals were extrapolated to the endwall using a linear attenuation profile with typical

attenuation rates of 0.8 to 1.5% per meter. In addition to the five pressure transducers

used for shock velocity an additional Kistler transducer (603B1 transducer and 5010B

amplifier) was used to measure the pressure time-history during shock-heating. The pre-

11

shock initial mixture pressure was measured using one of two high-accuracy MKS

Baratron 690A capacitance manometers (100 and 10,000 Torr models) with output given

by two MKS 270D signal conditioners. The incident and reflected shock conditions were

calculated using the normal shock equations; uncertainty in the experimental pressure and

temperature are ~1% and ~0.7%, respectively, with the primary contribution being the

uncertainty in the measured shock velocity; for details on the uncertainty analysis for

pressure and temperature see J.T. Herbon’s Ph.D. thesis [96]. For most of the

experiments presented here, vibrational equilibrium can be assumed immediately behind

the incident and reflected shock; the one exception, is the experiments involving CO2. In

these experiments the finite rate of V-T transfer plays an important role and is discussed

later in this chapter and in Chapter 5.

Mixtures were made in a turbo-pumped stainless-steel mixing tank (12-liter) with

an internal stirring system and were allowed to mix for at least four hours prior to

experiments. Research grade argon (99.999%) was used as the driven carrier gas. The gas

mixtures were introduced into the mixing tank through a high-purity gas mixing

manifold. The manifold allowed for introduction of both gases from high-pressure

cylinders and liquid chemicals with vapor pressure into the mixing tank. The gas

pressures were monitored with two high-accuracy MKS Baratron 690A capacitance

manometers (100 and 10,000 Torr models). The mixing tank pressure was monitored

with a Setra 280E capacitance pressure transducer (250 psia). The mixing tank and

mixing manifold were evacuated with a mechanical pump and Varian V-80

turbomolecular pump, providing pressures of 10-6 Torr and leak rates of 10-6 Torr/min

when evacuated overnight.

Optical measurements were made at a location 2 cm from the endwall in the case

of reflected shock experiments and a location of 29 cm from the endwall for incident

shock experiments. The shock-heated gases were accessed through 0.75” diameter flat

windows made of either UV fused silica or CaF2 flush mounted to the inner radius of the

shock tube.

12

2.2 CH3 ultraviolet laser absorption diagnostic Prior to the studies performed for the determination of rate coefficients for

reactions (1)-(5) a suitable diagnostic was needed for the quantitative detection of methyl

radicals (CH3), a product species of reactions (1)-(4). Methyl radicals are known to

absorb near 216 nm in the B2A1’ X2A2

’’ electronic system [97]. The distinctive features

of the spectra are the P+Q band peak and the R band peak located at 216.62 nm and

215.9 nm respectively at high-temperature [98]; see Figure 2.1 for the CH3 absorption

spectrum. The absorption at the P+Q band peak shows less variation with temperature

than the R band peak. Therefore, 216.62 nm (the P+Q peak) was chosen as the

wavelength of choice for detecting methyl at high temperatures.

2.2.1 Optical setup The optical setup used to generate 216.62 nm laser radiation is shown in Figure

2.2. Laser light at 433.24 nm (200-400 mW) was created by pumping a Coherent 699-05

ring dye laser, operating on stilbene 420 dye (50 psi dye jet pressure), with 7 W (all-lines

UV) from a Coherent Innova 25/7 Ar+ laser. The output of the dye laser was then

frequency-doubled in a Spectra-Physics Wavetrain external doubling cavity using an

angle-tuned beta barium borate (β-BaB2O4, BBO) crystal to create 216.62 nm laser

radiation with a linewidth of 40 MHz. The wavelength was monitored by passing a

portion of the dye laser output (undoubled) into a Burleigh WA-1000 Wavemeter

(uncertainty ±0.017 cm-1); mode quality was monitored by passing a portion through a

scanning interferometer. The doubled ultraviolet beam (2-3 mW at 216.62 nm) was

passed to the shock tube where it was split into two components: one passing through the

shock tube roughly 1-2 mm in diameter to be absorbed by CH3 (I), the other was detected

prior to absorption as a reference (I0). These two beams were detected using UV

enhanced S1722-02 Hamamatsu silicon photodiodes (risetime < 0.5 µs, 4.1 mm diameter)

installed in Thor Labs PDA55 detectors and recorded on a digital oscilloscope. The

intensity noise characteristics of the Ar+ pump output (all-lines UV), dye laser output

(433.24 nm), and frequency doubled output (216.62 nm) are given in Figure 2.3 and

Table 2.1.

13

Figure 2.3 demonstrates that the intensity noise in the Ar+ pump and dye laser are

comparable but the intensity noise in the 216.62 nm doubled output is substantially

greater than that of the 433.24 nm dye laser output. The noise amplification in the

harmonic output is due to frequency instability in the dye laser fundamental. The actively

stabilized Wavetrain doubling cavity can only track frequency changes that occur at less

than 10 GHz per second; instabilities that result in frequency changes that occur at faster

rate result in intensity noise in the harmonic output (216.62 nm).

The Coherent 699-05 ring dye laser was passively stabilized with two etalons

contained in the intra-cavity assembly (ICA): a 10 GHz thick etalon and a 225 GHz thin

etalon. These etalons caused an attenuation of about 40% of the broadband power but

ensured a single frequency within an effective 40 MHz linewidth (±20 MHz). The

instantaneous linewidth of the laser is a fraction of a hertz when operating in a single

mode but dye jet fluctuations broaden the linewidth resulting in this effective linewidth of

40 MHz. However, the rate of jitter of the instantaneous linewidth within the effective 40

MHz linewidth is not controllable and the frequency jitter induced intensity noise results

in the doubling cavity. A two-beam common-mode rejection scheme can be used to reject

the noise in the 216.62 nm laser beam providing acceptable signal-to-noise for shock tube

experiments in the resulting absorbance (minimum detection ~0.1% absorbance), see

Figure 2.4 and Table 2.1.

The detection of 216.62 nm laser radiation (and the other wavelengths discussed

later in this chapter) was carried out using modified Thorlabs PDA55 detectors. The

detectors are modified by replacing the stock 1 mm diameter Thorlabs silicon photodiode

with a UV-enhanced Hamamatsu S1722-02 4.1 mm in diameter. The UV-enhanced

photodiode allows for greater UV sensitivity down to 200 nm without sacrificing time

response. The PDA55 detectors are easily modified by unsoldering the stock photodiodes

and soldering in the Hamamatsu photodiodes.

Unfortunately deep UV light damages silicon [199]. Therefore, we were careful in

the manner in which we detected UV laser radiation (216, 244, and 266 nm). After

exposure to focused laser radiation the photodiodes exhibit reduced sensitivity to UV

light at the location where the laser beam was focused. In fact a sensitivity hole in the

detector at the location of the focused laser beam occurs, as shown in Figure 2.5 for 266

14

nm light. The reduced sensitivity at the beam location can manifest itself as beam

steering noise in shock wave experiments. To avoid burning sensitivity holes in the

detectors and to avoid beam steering fluctuations it was determined the best solution for

detection is to expand the beam onto the detector using a lens so that it covers the

majority of the 4.1 mm diameter photodiode surface. Additionally, one must be very

careful to never leave the UV beam incident on the detector unless aligning or performing

an experiment.

2.2.2 CH3 absorption coefficient measurements and results For determination of the methyl absorption coefficient at 216.62 nm, methyl

radicals were generated immediately behind shock waves by the dissociation of a

precursor. Three different precursors were used for various temperature ranges:

azomethane (CH3N2CH3) for the lowest temperatures (1219-1687 K), methyl iodide

(CH3I) for intermediate temperatures (1733-2146 K), and ethane (C2H6) for the highest

temperatures (1938-2501 K). Low concentrations of these precursors (200 or 400 ppm)

were mixed with argon prior to experiments. Azomethane is not sold commercially and

was synthesized using the method of Jahn [99] which has previously been found to be

cleaner and easier than the method of Renaud and Leitch [100]; see Appendix D for

synthesis instructions. The CH3 precursors were chosen for a given experiment such that

they would dissociate quickly behind the shock wave. In the case of experiments

performed behind reflected shock waves, care was taken not to dissociate the precursor

behind the incident shock wave, as this would lead to the formation of ethane, via

reaction (10) below, which would then slowly dissociate behind the reflected shock wave.

This problem necessitated the use of incident shock heating for some experiments.

In a given experiment, regardless of the precursor, the CH3 profiles had the same

temporal characteristics. At time zero the shock wave (incident or reflected depending on

the experiment) passed the laser beam causing a schlieren (density deflection) spike in

the signal. Following shock heating the CH3 absorbance (-ln(I / I0)) rises immediately to a

peak due to the decomposition of the precursor and then decays primarily due to methyl-

methyl reactions:

CH3 + CH3 → C2H6 (10)

15

CH3 + CH3 → C2H5 + H (11)

The absorbance traces allow the extraction of the absorption coefficient at 216.62 nm by

back-extrapolating the early time decay in absorbance to time-zero and using Beer’s law

I / I0 = exp(-kλ,CH3 P XCH3 L)

where I is the transmitted laser intensity, I0 is the reference beam intensity, kλ,CH3

[atm-1cm-1] is the temperature-dependent CH3 absorption coefficient at 216.62 nm, P

[atm] is the total pressure, XCH3 is the CH3 mole fraction, and L is the absorption path

length (diameter of the shock tube 14.13 cm). Additionally, the total rate coefficient for

CH3 + CH3, k10 + k11, can be fit to the decay in absorbance. See Figure 2.6 for an example

experiment and Figure 2.7 for the absorption coefficient results with comparison to

previous studies. The uncertainty in the absorption coefficient obtained by these

experiments, estimated to be ±5%, is the result of uncertainties in experimental

temperature and pressure, uncertainties in the initial concentration of methyl precursors,

and uncertainties associated with the fitting of experimental traces. A least-squares

expression for the absorption coefficient is given by

kλ,CH3 = 1.475 x 104 T -1.004 exp(2109K/ T) [atm-1 cm-1]

Additionally, no pressure-dependence in the CH3 absorption coefficient was detectable in

experiments up to 8 atm.

The current absorption coefficient results are in good agreement with the previous

laser absorption results from our laboratory, Davidson et al. [101], but the scatter in the

data is only ±5%, at least a factor of two smaller than the scatter in the data of Davidson

et al. The broadband lamp measurements of Glänzer et al. [102] and Möller et al. [103]

are in good agreement with the current determination, while the broadband lamp

determination of Tsuboi [104] lies lower than the current determination. The narrow line-

width measurements of Hwang et al. [105] (213.9 nm) and Du et al. [106] (215.94 nm)

differ from the current findings due to the different wavelengths used in these studies.

2.3 CO2 ultraviolet laser absorption diagnostic Ultraviolet absorption due to vibrationally hot CO2 was used to investigate the

incubation and decomposition of CO2. In addition the ultraviolet absorption of hot CO2

provides opportunity for thermometry measurements in high-temperature gases.

16

Appendix C describes thermometry measurements behind shock waves in reacting and

non-reacting mixtures similar to the measurements Jeffries et al. [107] and Mattison et al.

[108] have made in pulse detonation engines and reciprocating internal combustion

engines. Measurement of absorption cross-sections were made at four available laser

wavelengths: 216.5, 244, 266, and 306 nm.

2.3.1 Optical setup Continuous-wave laser radiation was generated at four different wavelengths:

216.5 nm (2-3 mW) was generated by doubling the output of an Ar+ pumped dye-laser,

operating at 433 nm, in a BBO external frequency-doubling cavity, described above in

section 2.2.1; 306 nm (2-3 mW) was generated by intra-cavity frequency doubling in a

Nd:YVO4-pumped (532 nm) dye laser operating at 612 nm, as described in previous OH

radical studies by Herbon and co-workers [96]; and 244 and 266 nm (1.5 mW each) were

produced by the single pass of a focused laser beam at 488 nm (Ar+ line at 7 W) and 532

nm (Nd:YVO4 at 5 W) through angle-tuned BBO. The harmonics (244 and 266 nm) were

separated from the fundamental beams (488 and 532 nm) with a set of Pellin-Broca

prisms. See Figure 2.8 for a schematic of the optical setup used to generate 244 and 266

nm laser radiation. The intensity noise characteristics of the 244 and 266 nm output are

shown in Figure 2.9 and Table 2.1. Again a two beam common-mode rejection scheme

was used to reject the noise in the 244 nm signal providing acceptable signal-to-noise for

shock tube experiments in the resulting absorbance (minimum detection ~0.1%

absorbance).

The UV laser beams were split into two components: one, ~1 mm in diameter,

passing through the shock tube to be absorbed by CO2 (I), and one detected prior to

absorption as a reference (I0). These two beams were detected using silicon photodiodes

(Thor Labs PDA55 detector and Hamamatsu S1722-02 diode) and recorded on a digital

oscilloscope. The absorption cross-sections were determined using Beer’s law

I / I0 = exp(-σCO2(λ,T) nCO2 L)

where I is the transmitted laser intensity, I0 is the reference beam intensity, σCO2(λ,T)

[cm2molecule-1] is the wavelength- and temperature-dependent CO2 absorption cross-

section, nCO2 [molecules cm-3] is the CO2 number density, and L is the absorption path

17

length (diameter of the shock tube 14.13 cm). For these experiments absorption cross-

sections were not derived for experiments in which the absorbance was less than 0.7%.

2.3.2 CO2 ultraviolet absorption cross-section measurements The absorbance (ln(I0 / I)) traces for several experiments are shown in Figure

2.10. The passage of the incident and reflected shock waves cause two schlieren spikes in

the signal, due to steering of the beam off the detector. After the passage of the reflected

shock (marked by a strong schlieren induced signal), there is a very short risetime in the

absorption signal (~1-5 µs depending on the temperature and pressure) prior to the steady

absorption due to hot CO2. The absorption cross-section is determined during this

plateau. This short induction time is caused by the CO2/Ar V-T relaxation process [109-

110] which brings the vibrationally cold CO2 to an excited vibrational condition where

absorption is possible. The observation of this induction time provides evidence that the

high-temperature UV absorption of CO2 comes from a vibrationally-excited CO2 ground

state (1Σg+). It has been shown by Spielfiedel et al. [111] that molecules with excited

bending modes could account for the hot CO2 absorption at wavelengths longer than 200

nm because of the significantly greater Franck-Condon overlap these states have with the

bent CO2(1B2) electronically excited upper state. The data shown in Figure 2.10 are at

temperatures below 3000 K and do not show significant thermal decomposition in the

first 200 µs. Note, also that the incident shock conditions also reach the vibrationally

equilibrated conditions within the incident shock-heating time period for all experiments

presented here.

At higher temperatures (>3000 K) the CO2 undergoes thermal decomposition and

the absorption traces show temporal decay at long times behind the reflected shock; an

example experiment is shown in Figure 2.11. This example shows a significant induction

prior to constant absorption following the incident shock due to vibrational relaxation; the

vibrational equilibrated conditions behind the incident shock are 1812 K and 0.161 atm.

Although, the vibrational relaxation is resolved after the incident shock, it is not resolved

after the reflected shock because the higher temperature and pressure behind the reflected

wave have a much faster vibrational energy transfer rate. Following the passage of the

reflected shock wave an incubation period is observed in the absorption prior to the

18

thermal decomposition. The reflected shock conditions are 3838 K and 0.820 atm for the

experiment in Figure 2.11. To our knowledge this is the first observation of incubation

proceeding decomposition in CO2, incubation is discussed in detail in Chapter 5. The

absorption cross-section was determined for these decomposition experiments in the

incubation plateau prior to dissociation, and therefore, the cross-section determination

does not depend on knowledge of the subsequent chemistry. Additionally, the

experimental temperature during the plateau is not in question for these high-temperature

experiments because the CO2 vibrational temperature equilibrates to the translational

temperature within ~1 µs (the vibrational relaxation time in these high-temperature

decomposition experiments).

2.3.3 CO2 ultraviolet absorption cross-section results The measured absorption cross-sections at the four wavelengths are plotted versus

temperature in Figure 2.12 and compared with the data from the earlier study from our

laboratory [112]. The agreement between the two studies is excellent for temperatures

between 1800 to 2800 K; at higher temperatures, the extrapolation from the earlier lower

temperature data would predict too large a cross section.

Due to the spectrally smooth nature of the UV CO2 absorption feature [112] the

current absorption cross-section results can be fit to a semi-empirical form

ln σCO2(λ, T) = a + bλ

where a = c1 + c2T + c3/T and b = d1 + d2T + d3/T. The cross-section, σCO2(λ, T), is in

units of 10-19 cm2molecule-1, the wavelength, λ, is in units of 100 nm, and the

temperature, T, is given in units of 1000 K. The following parameters provide a best fit to

the data: c1 = 0.05449, c2 = 0.13766, c3 = 23.529, d1 = 1.991, d2 = -0.17125, and d3 = -

14.694. As is shown in Figure 2.12 the scatter of the data about the fit is quite small with

a 1-σ standard deviation of ±1.8%. It should be noted that this semi-empirical expression

for the absorption cross-section should not be extrapolated outside the temperature and

wavelength range of the data, but based on experience from the Schulz et al. [112] study

interpolation is justified.

The uncertainty in the CO2 cross-sections obtained in these experiments is the

result of uncertainties in the experimental temperature and pressure, uncertainties in the

19

initial CO2 concentration, and uncertainties associated with the signal-to-noise of a given

experiment. The experiments in which the smallest absorption was monitored (low T and

long λ) provide cross-sections with the highest uncertainty. The majority of the cross-

section results have uncertainties less than 5%; only a few of the smallest absorption

cross-sections measured at 244 and 266 nm have larger uncertainties, with the largest

uncertainty of 14%.

A comparison of the new cross-section determinations with those of Schulz et al.

[112] (Figures 2.11 and 2.12) shows that the agreement is quite good in the middle of the

temperature range, with deviations of 30% and 40% respectively at the highest and

lowest temperatures of the Schulz et al. study which is within the reported uncertainty for

the Schulz et al. data at the low- and high-temperature extremes. At low-temperatures the

difference in the current findings and Schulz et al. is likely due to limited signal-to-noise

in the Schulz et al. experiment; as the noise in the spectra (Figure 2.13) shows, the Schulz

et al. results at low-temperatures and long wavelengths are significantly more uncertain

than the current results due to poorer signal-to-noise. At high temperatures the current

absorption cross-sections are lower than those of Schulz et al. due to their use of data

impacted by CO2 thermal decomposition. The current experiments indicate that the

mechanism used in the Schulz et al. study (GRI Mech 3.0 [95]) to account for

decomposition by CO2 has a rate coefficient for CO2 → CO + O that is too fast. Schulz et

al. calculated the average mole fraction of CO2 during their averaging time period using

the GRI mechanism, thus giving a low CO2 mole fraction. Thus they inferred an

absorption cross-section that is too large. The current cross-section results do not rely on

a chemical model to account for decomposition at high-temperature due to the

microsecond time resolution provided by laser absorption.

2.3.4 CO2 ultraviolet absorption ground state energy levels The hot CO2 pre-dissociative absorption is a continuum spectral feature caused by

the intersection of many electronic-vibrational-rotational lines, with a dense energy

spectrum typical of a triatomic. If one uses a quasi-diatomic approximation to describe

these optical transitions [113], it can be shown that the absorbance at a given wavelength,

λ, is described by

20

σCO2(λ,T)n = σCO2,eff(λ)n1 = [σCO2,eff(λ)n / Qvib] exp(-ε1(λ) / kT)

where the average absorption takes place from a population with density n1 at a lower

state vibrational energy ε1(λ) above the zero-energy ground state and where σCO2,eff(λ) is

an effective cross-section for absorption. The vibrational partition function, Qvib, is given

by

Qvib = [1 – exp(-θ1/T)]-1 [1 – exp(-θ2/T)]-2 [1 – exp(-θ3/T)]-1

where θ1 = 1999 K, θ2 = 960 K, and θ3 = 3383 K. This description of the temperature-

dependent absorption cross-section allows estimation of the effective vibrational energy

of the absorbing lower state, ε1(λ). The top graph in Figure 2.14 shows a plot of the

product of the measured cross-section, σCO2(λ,T), with the vibrational partition function,

Qvib versus inverse temperature. This plot shows the product, σCO2(λ,T)Qvib, is linear with

a slope that provides an estimate of the absorbing lower state energy, ε1. The cross-

section measurements provide lower state energies of 1.09 eV (8790 cm-1), 1.48 eV

(11940 cm-1), 1.79 eV (14440 cm-1), and 2.34 eV (18870 cm-1) for absorption of photons

at 216.5, 244, 266, and 306 nm respectively; these results can be expressed with

ε1(λ) = -1.94 + 0.014 λ [eV]

where the wavelength, λ, is in nm. The lower state energy results are compared to

previous determinations made by Eremin et al. [114] (193 nm) and Zabelinskii et al.

[115] (238 and 300 nm) in the bottom graph of Figure 2.14 with good agreement.

Unfortunately, this simplified photophysical model does not allow for an assignment of

the absorption at a given wavelength to a specific vibrational level(s) because of the

complex nature of the triatomic energy spectrum. Also, note that the linearity of

σCO2(λ,T)Qvib deviates slightly at the highest temperatures (top graph Figure 2.14). This

deviation might be caused by enhanced absorption due to transitions to electronic states

other than 1B2 such as 1A2, 3A2, and 3B2 that occurs at the highest temperatures where

there is more highly excited population.

21

2.4 Benzyl radical ultraviolet laser absorption

diagnostic Benzyl radicals have strong broadband absorption in the ultraviolet from 245 to

275 nm [82,116] (see Figure 2.15) allowing detection of benzyl at 266 nm using the

technique described in section 2.3.1 for generation of 266 nm laser radiation. Shock wave

experiments were performed using the fast decomposition of benzyl iodide as a benzyl

source. The experiments allowed determinations of the absorption cross-section of benzyl

at 266 nm and the rate coefficient for benzyl decomposition, reaction (7). Here the

absorption cross-section results are discussed and in Chapter 6 the decomposition rate

coefficient is discussed.

2.4.1 Benzyl radical absorption cross-section measurements

and results Benzyl radicals (C6H5CH2) were generated immediately behind shock waves via

the dissociation of benzyl iodide

C6H5CH2I → C6H5CH2 + I (12)

Benzyl iodide, supplied by the Narchem Corporation (Chicago, IL) at 99% purity, was

mixed with argon in the mixing tank at concentrations of 50 and 100 ppm prior to

experiments. Benzyl iodide has a fusion temperature of 24.5º C at 1 atm. Therefore, for

the successful introduction of benzyl iodide into the mixing tank the room temperature

was raised to 26º C to increase the benzyl iodide vapor pressure and the tank was heated

to 35º C and 45º C using a resistance heating tape to avoid wall condensation. The two

sets of experiments (mixing tank at 35º C and 45º C) resulted in identical absorption

cross-section results providing confidence that the benzyl iodide concentration in the

mixtures was accurate as determined manometrically. Additionally, the benzyl iodide was

cycled through several freeze-pump-thaw cycles prior to introduction into the mixing

tank to avoid impurities from high volatility species. Mixtures were allowed to mix

overnight prior to shock wave experiments to allow for complete mixing.

22

The absorbance at 266 nm for an example experiment is shown in Figure 2.16.

Prior to the incident shock wave, no 266 nm absorption by benzyl iodide was detected at

the low initial pressure, P1. The passing of the incident shock wave causes a schlieren

spike in the signal and the test gas is compressed and heated causing the benzyl iodide to

absorb (1% absorbance at t ≈ 60 µs in Figure 2.16). Behind the incident shock wave the

benzyl iodide (C6H5CH2-I bond strength 51 kcal/mol) slowly dissociates into the more

strongly absorbing benzyl radical and an I-atom. While the incident shock temperature is

not sufficient to induce benzyl radical decomposition, the benzyl can combine to give

stable dibenzyl

C6H5CH2 + C6H5CH2 → C6H5CH2CH2C6H5 (13)

The passage of the reflected shock wave causes complete dissociation of the benzyl

iodide and the small amount of dibenzyl (Bz-Bz bond strength 65.2 kcal/mol) that was

formed in the incident test time yielding one benzyl per benzyl iodide in the test mixture.

The absorbance at time-zero after the reflected shock wave passing gives the 266 nm

absorption cross-section of the benzyl radical. The absorbance at time-zero was found by

back-extrapolating the steady decay in the absorbance signal to time-zero, defined as the

center of the schlieren spike.

The decay in the absorbance signal after the passing of the reflected shock wave is

primarily due to the decomposition of benzyl yielding a more weakly absorbing C7H6

molecule of unknown structure and an H-atom [117]

C6H5CH2 → C7H6 + H (7)

To determine the rate coefficient for benzyl decomposition, reaction (7), the decay in

absorbance was fit by using a detailed kinetic mechanism and accounting for the

interfering absorption due to benzyl fragments and toluene. Toluene is formed by the

secondary reaction

C6H5CH2 +H → C6H5CH3 (14)

and decays by reactions (8) and (9). Chapter 6 contains discussion of the fit of the benzyl

decay for determination of k7 and the study of toluene decomposition, reactions (8) and

(9).

The absorbance at long times (Figure 2.16) decays to approximately a constant

value. Assuming a complete one-to-one conversion of benzyl to benzyl fragments at long

23

times, the absorption cross-section of the benzyl fragments was determined. The

absorption cross-section of the benzyl fragments is needed for the rate coefficient studies

presented in Chapter 6 where benzyl fragment interfering absorption must be accounted

for.

The results for the absorption cross-section of benzyl radicals and benzyl

fragments are given in Figures 2.16 and 2.17 with comparison to previous experimental

studies. The benzyl radical and benzyl fragment absorption cross-sections showed no

temperature-dependence from 1428 to 1730 K with values of σC6H5CH2 = 1.9 (±0.19) x

10-17 cm2molecule-1 and σC7H6 = 3.4 (±0.51) x 10-18 cm2molecule-1 respectively. The

results for the benzyl radical absorption cross-section are in good agreement with the

flash photolysis measurement of Ikeda et al. [116] but are about 50% lower than the

shock tube results of Müller-Markgraf and Troe [82], see Figure 2.17. Müller-Markgraf

and Troe examined the absorption spectra of benzyl behind shock waves at 1600 K using

a Xe high-pressure discharge lamp dispersed with a monochromator. In comparison to

the laser absorption measurements presented here, the Müller-Markgraf and Troe lamp

measurements have noise at least a factor of ten larger due to the limited spectral

brightness and optical collection efficiency of their lamp absorption measurements.

Therefore, the resulting absorption cross-section measured by Müller-Markgraf and Troe

is significantly less certain than the current measurement. However, the results of the

current benzyl fragment absorption cross-section measurements are in excellent

agreement with the measurements of Müller-Markgraf and Troe (Figure 2.18).

24

Table 2.1: Laser RMS intensity noise for 1 ms measurement. Optical component RMS Intensity noise over 1 ms Detector dark noise (Thor Labs PDA55 detector with UV enhanced S1722-02 Hamamatsu silicon photodiodes)

0.56 mV

216.62 nm Ar+ pump (all-lines UV) ±0.098%

699-05 dye laser (433.24 nm) ±0.15%

Doubling cavity output (216.62 nm) ±0.78%

216.62 nm absorbance after common-mode rejection ±0.09%

244 nm

244 nm doubled Ar+ line ±0.13%

244 nm absorbance after common-mode rejection ±0.09%

266 nm 266 nm doubled Nd:YVO4 ±0.06%

266 nm absorbance after common-mode rejection ±0.06% (common-mode rejection

does not improve noise for 1 ms duration but does cancel very slow intensity fluctuations)

25

211 212 213 214 215 216 217 218 219 220 221

0

10

20

30

40

k CH

3 [cm

-1 a

tm-1]

Wavelength [nm]

P+Q PeakR Peak

Figure 2.1: Absorption spectra of CH3 B2A1

’ X2A2’’ electronic system at 1565 K,

Oehlschlaeger et al. [118].

Figure 2.2: Schematic of 216.62 nm experimental setup: M, mirror; BS, beam splitter; L, lens; D, detector; PT, piezoelectric pressure transducer; and Kistler, shielded Kistler piezoelectric pressure transducer.

26

0.0 0.2 0.4 0.6 0.8 1.0-3

-2

-1

0

1

2

3 -0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.8 1.0

-0.4-0.3-0.2-0.10.00.10.20.30.4

Time [ms]

doubling cavity (216.62 nm)

In

tens

ity N

oise

[% o

f sig

nal]

dye laser (433.24 nm)

Time [ms]

Ar+ pump (all-lines UV)

Figure 2.3: Laser intensity noise measured with Thor Labs PDA55 detector and Hamamatsu S1722-02 silicon photodiode. Top graph, Ar+ pump output (all-lines UV), RMS noise is ±0.098%; middle graph, dye laser output (433.24 nm), RMS noise is ±0.15%; bottom graph, doubling cavity output (216.62 nm), RMS noise is ±0.78%.

27

0.0 0.2 0.4 0.6 0.8 1.0-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

Abs

orba

nce

Time [ms]

Figure 2.4: Absorbance signal at 216.62 nm using two-beam common-mode rejection with Thor Labs PDA55 detectors and Hamamatsu S1722-02 silicon photodiodes, RMS noise is ±0.09%.

-1 0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Det

ecto

r Sen

sitiv

ity @

266

nm

Distance [mm]

Figure 2.5: Modified Thorlabs PDA55 (Hamamatsu S1722-02 silicon photodiode) detector spatial sensitivity at 266 nm after a focused 1 mW 266 nm beam was incident on the photodiode surface.

28

-100 -50 0 50 100 150 200

0

100

200

300

400

CH

3 Mol

e Fr

actio

n [p

pm]

Time [µs]

Figure 2.6: Example azomethane decomposition data and modeling. Reflected shock conditions: 1561 K, 1.622 atm, 200 ppm azomethane/Ar; laser wavelength = 216.62 nm, kλ,CH3 = 34.3 cm-1atm-1.

1200 1400 1600 1800 2000 2200 240010

20

30

40

50

60708090

100

k CH

3 [cm

-1 a

tm-1]

Temperature [K]

Figure 2.7: Temperature dependent CH3 absorption coefficient at 216.62 nm. Solid squares, azomethane data; solid circles, methyl iodide data; solid triangles, ethane data; solid line, fit from this study; x, Davidson et al. [101]; open circles, Tsuboi [104]; dashed line, Möller et al. [103]; dotted line, Glänzer et al. [102]; dash-dot line Du et al. [106] (215.94 nm); dash-dot-dot line, Hwang et al. [105] (213.9 nm).

29

Figure 2.8: Schematic of 244 and 266 nm experimental setup: L, lens; BBO, beta barium borate frequency doubling crystal; PB, Pellin-Broca prism; M, mirror; BS, beam splitter; D, detector; PT, piezoelectric pressure transducer; and Kistler, shielded Kister piezoelectric pressure transducer.

30

0.0 0.2 0.4 0.6 0.8 1.0-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.8 1.0

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

266 nm

Time [ms]

Time [ms]

Inte

nsity

Noi

se [%

of s

igna

l]

244 nm

Figure 2.9: Laser intensity noise measured with Thor Labs PDA55 detector and Hamamatsu S1722-02 silicon photodiode. Top graph, 244 nm output (doubled 488 nm Ar+ line), RMS noise is ±0.13%; bottom graph, 266 nm output (doubled 532 nm Nd:YVO4), RMS noise is ±0.06%.

31

-50 0 50 100 150 200

0.00

0.05

0.10

0.15

0.20

266 nm, 2969 K, 0.671 atm, 2% CO2 / Ar

306 nm, 2919 K, 0.990 atm, 10% CO2 / Ar

244 nm, 2782 K, 1.097 atm, 2% CO2 / Ar

Abs

orba

nce

Time [µs]

216.5 nm, 2871 K, 1.067 atm, 2% CO2 / Ar

Figure 2.10: Example CO2 absorbance for experiments at 216.5, 244, 266, and 306 nm.

-50 0 50 100

0.00

0.05

0.10

0.15

0.20

Abs

orba

nce

Time [µs]

Incubation Time

VibrationalRelaxation

Figure 2.11: Example CO2 absorbance at 216.5 nm for experiment with thermal decomposition. Initial mixture: 2% CO2 / Ar. Vibrationally equilibrated incident shock conditions: 1812 K and 0.161 atm. Initial (prior to decomposition) vibrationally equilibrated reflected shock conditions: 3838 K, 0.820 atm. Note the vibrational relaxation after incident shock-heating and the incubation period prior to decomposition after reflected shock-heating.

32

2000 3000 400010-21

10-20

10-19

10-18

306 nm

266 nm

244 nm

σ CO

2 [cm

2 mol

ecul

e-1]

Temperature [K]

216.5 nm

Figure 2.12: Temperature-dependence of the UV CO2 absorption cross-section. Solid symbols, experimental results: squares, 216.5 nm; circles, 244 nm; triangles, 266 nm; diamonds, 306 nm. Solid lines, semi-empirical fit to current data; dashed lines, Schulz et al. [112].

200 220 240 260 280 300

10-21

10-20

10-19

10-18

1630 K2010 K

2310 K

2610 K

Wavelength [nm]

σ CO

2 [c

m2 m

olec

ule-1

]

3050 K

Figure 2.13: Wavelength-dependence of the UV CO2 absorption cross-section. Solid symbols and line, current data: squares, 3050 K; circles, 2610 K, diamonds, 2010 K; stars, 1630 K. Dashed lines, Schulz et al. [112].

33

200 220 240 260 280 300 3200.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Low

er S

tate

Ene

rgy

[eV

]

Wavelength [nm]

Oehlschlaeger (2004) Eremin et al. (1998) Zabelinskii et al. (1985)

0.2 0.3 0.4 0.5 0.6 0.7

10-20

10-19

10-18

10-17

306 nm

266 nm

244 nm

1000/T [1/K]

σ CO

2Qvi

b [cm

2 mol

ecul

e-1]

216.5 nm

Figure 2.14: Product of cross-section and vibrational partition function (σCO2Qvib) versus inverse temperature (top graph) and ground state energy versus wavelength (bottom graph). Top graph: solid squares, 216.5 nm data; solid circles, 244 nm data; solid triangles, 266 nm data; solid diamonds, 306 nm data; solid lines, least-squares fit of form σCO2(λ,T)Qvib = σCO2,eff(λ)exp(-ε1/T). Bottom graph: squares, current results; line, linear fit to current results; open circles, Eremin et al. [114]; ×, Zabelinskii et al. [115].

34

180 200 220 240 260 280 300 32010-18

10-17

10-16

σ C6H

5CH

2 [cm

2 mol

ecul

e-1]

Wavelength [nm]

Figure 2.15: Benzyl radical spectrum at 1600 K, Müller-Markgraf and Troe [82].

-100 0 100 200 300 400 500 600 700 800

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Abs

orba

nce

Time [µs]

Figure 2.16: Example 266 nm absorbance during benzyl iodide decomposition. Reflected shock conditions: 1615 K, 1.556 atm, 50 ppm benzyl iodide/Ar.

35

1400 1500 1600 1700 18001

2

3

4

5

σ C6H

5CH

2 [1

0-17 cm

2 mol

ecul

e-1]

Temperature [K]

Figure 2.17: Benzyl radical absorption cross-section at 266 nm. Solid squares, this study; open circle, Ikeda et al. [116]; open triangle, Müller-Markgraf and Troe [82].

1400 1500 1600 1700 18001

2

3

4

5

6

789

10

σ C7H

6 [1

0-18 cm

2 mol

ecul

e-1]

Temperature [K]

Figure 2.18: Benzyl fragment absorption cross-section at 266 nm. Solid squares, this study; open triangle, Müller-Markgraf and Troe [82].

36

Chapter 3: Theory and computational

techniques

3.1 Unimolecular theory Many reactions, such as dissociation, association, and isomerization reactions,

experimentally exhibit pressure-dependence as well as temperature-dependence.

Lindemann first proposed a mechanism in 1922 to explain this phenomenon [119]. He

proposed that these reactions take place in two steps:

MAMAe

de

k

k+⇔+ * (e, de)

PAunik

→* (uni)

The molecule A is energized by collision with collider M in reaction (e). Molecule A then

can be deactivated by collision, reaction (de), or can proceed to products by reaction

(uni). Applying a steady-state approximation for the population of A* the rate of reaction

can be written

unide

e

kMkMAk

dtPd

/][1]][[][

+=

and the rate coefficient in the high-pressure-limit ([M] → ∞) and low-pressure-limit ([M]

→ 0) can be defined respectively as

][0 Mkk

kkk

k

e

de

euni

=

=∞

At intermediate pressures, the so-called falloff regime, the Lindemann apparent first

order rate coefficient k1 can be defined as

LFkk

kkkk

=+

=∞

∞

∞ /1/

0

01

37

The Lindemann mechanism demonstrates that the apparent first-order rate

coefficient is limited by collisional excitation at low-pressure and the unimolecular

conversion at high-pressure. However, experimental findings have shown that typically

the falloff is much “broader” than that given by the Lindemann model. In other words the

value [M]1/2 = k0/k∞ defined as the concentration when k1/k∞ = 1/2 is found

experimentally to be smaller than that predicted by the Lindemann model. To better

describe the experimentally observed falloff the energy-dependence of the activation and

unimolecular steps must be considered.

The modern technique commonly used to calculate the energy dependence of

unimolecular reaction rate coefficients is described by Rice-Ramsperger-Kassel-Marcus

(RRKM) theory [120]. According to RRKM theory the energy-specific unimolecular rate

coefficient k(E) is given by

)()(

)( 0‡

‡

EhEEW

lEkρ−

=

where l‡ is the reaction path degeneracy, W‡(E - E0) the sum of states of the transition

state, ρ(E) the density of states of the reactant molecule, and h Planck’s constant. The

energy E is measured relative to the zero-point energy of the reactant molecule and the

threshold energy E0 is the difference between the zero-point energies of the reactant and

transition state.

The competition between excitation and unimolecular conversion results in

different population distributions at low- and high-pressure. At high-pressure the

unimolecular conversion is rate-limiting and an equilibrium Boltzmann distribution is

maintained through collisions. However, at low-pressure unimolecular conversion occurs

faster than collisional activation and the population is depleted above the reaction

threshold. See Figure 3.1 for a schematic representation of the population distribution in

these two regimes. Because the population maintains a Boltzmann distribution at infinite

pressure the high-pressure rate coefficient can be calculated by averaging the RRKM

result for k(E) over the Boltzmann population f(E):

38

∫∫

∫ ∞

∞−

∞−

∞ ==0

0

/

/

)()()(

)()(0 dEEfEk

dEeE

dEeEEkk

kTE

E

kTE

ρ

ρ

where

QeEEf

kTE /)()(−

=ρ and Q is the total partition function.

With a simple expression for the high-pressure-limit rate coefficient the challenge

of modern unimolecular theory has been to describe the falloff to the low-pressure-limit.

In the falloff regime, collisional activation plays a role in determining the magnitude of

the rate coefficient; therefore, a method is needed to describe collisional energy transfer.

Several methods have been developed; in this work the master equation for internal

energy is solved numerically to describe energy transfer following the work of Barker

and others [123-124].

The master equation can be written by considering the time evolution of the

population, ρi, of the reactant molecule at a given energy level i simultaneously

undergoing unimolecular reaction and collisional activation and deactivation

( ) iij

ijijiji kPP

tρρρω

ρ−−=

∂∂ ∑

where ω is the inelastic collision frequency, ki the energy-specific rate coefficient from

RRKM theory, Pij the probability that a molecule in energy level j will undergo

collisional transition into level i, and Pji the probability that a molecule in energy level i

will undergo collisional transition into level j. The first term in the master-equation

represents the rate of population growth in level i due to collisional transfer into the level,

the second term represents the rate of population decay in level i due to collisional

transfer out of level i, and the last term represents the rate of population leaving level i

due to unimolecular reaction (dissociation in this thesis). It should be noted that the one-

dimensional master equation formulation, given above, neglects the population

dependence on angular momentum by assuming that all rotational degrees of freedom are

in thermal equilibrium (i.e., rotational energy transfer is much faster than vibrational

energy transfer). The master equation has been solved analytically for some special cases

39

[57-58, 121-122], usually involving small molecules, but for practical systems the master

equation must be solved numerically by discretizing the energy domain of interest.

3.2 Master equation / RRKM computational techniques In the studies presented here master equation simulations have been performed

using the MultiWell code developed by Barker [123-124]. MultiWell performs stochastic

numerical solution to the energy grained master equation. Additionally, MultiWell allows

for master equation solution to reaction systems involving multiple channels, for example

in this thesis n-butane decomposition, reactions (4)-(5), and toluene decomposition,

reactions (8)-(9). In MultiWell the energy-specific rate coefficient can be calculated using

one of two methods: 1) RRKM theory, described above or 2) the inverse Laplace

transform method.

For calculations in which RRKM theory is used to calculate k(E) the density of

states of the reactant molecule and sum of states of the transition state must be calculated.

This can be done using DenSum, an auxiliary code to MultiWell. DenSum uses the Stein-

Rabinovitch [127] version of the Beyer-Swinehart algorithm [128] for exact counts of

states of the molecule and allows the degrees of freedom of the molecule to be input as

harmonic oscillators, Morse oscillators, or free rotors. RRKM theory requires a model for

the transition state in order to calculate its sum of states. Many techniques for modeling

the transition state of dissociation reactions have been developed but in this work the

simple restricted Gorin model will be used [120,129-130]. The transition state was

located at the maximum of an effective potential, using the Lennard-Jones form of the

potential: r‡/r = (6D0/RT)1/6. The restricted Gorin transition state is similar to its product

fragments. The transition state frequencies of vibration are assumed to be those of the

product fragments and internal rotations are represented by the moments of inertia of the

product fragments with hindrance, η, chosen to reproduce experimental data.

Additionally, in this work the externally active k-rotor was included in the density and

sum of states calculations.

An alternative to RRKM theory for calculation of the energy-specific rate

coefficient k(E) is the inverse Laplace transform method given by Forst [125-126]

40

)()(

)(E

EEAEk

ρρ ∞

∞

−=

where A∞ and E∞ are the Arrhenius parameters for the high-pressure-limit rate coefficient

and ρ is the density of states for the parent dissociating molecule; in this formulation the

reaction path degeneracy has been included in A∞. The inverse Laplace transform method

is useful when the pressure of interest is nearly in the low-pressure-limit where the rate

coefficient is rate-limited by collisional activation and therefore, accurate calculation of

k(E) is not necessary. In this thesis the inverse Laplace transform method is used in

simulations of CO2 incubation and decomposition because at the conditions of the

experiments reported here the decomposition is nearly in the low-pressure-limit [44].

The last major input to MultiWell to be discussed here is the collisional energy

transfer probability density function, Pij. MultiWell has ten choices for the functional

form of the collisional energy transfer probability density function; the large number of

choices is due to the substantial research and debate on this subject. For the work

presented in this thesis, the exponential-down model is used for alkane decomposition

and CO2 incubation and decomposition

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

j

ij

jij

EEN

Pα

exp1 for Ej > Ei

where Nj is a normalization factor and αj is equal to the average energy transferred in

deactivating collisions, <∆E>down, when the exponential-down model is employed [120].

The collisional energy transfer parameter, αj, is fit to the experimental data in the falloff

and can have energy- and/or temperature-dependence. In the case of toluene

decomposition accurate measurements of Pij have been made by Luther and co-workers

[131-132]. They have chosen to represent their results in the following form which was

also adopted for the toluene master equation calculations performed in this work:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

γ

α j

ij

jij

EEN

P exp1 for Ej > Ei

The additional term γ allows for enhanced or decreased probabilities in the wings of the

distribution.

41

In general there are two adjustable parameters for every calculation: 1) the

hindrance parameter applied to the Gorin transition state internal rotations, η, and 2) the

energy transfer parameter α. The hindrance parameter, η, affects k(E) and therefore the

high-pressure-limit. The value of the energy transfer parameter, α, controls the rate of the

falloff to the low-pressure-limit. The specifics of each master equation calculation are left

to the results sections: Chapters 4-6.

3.3 Troe formulation for falloff As stated above, rate coefficients determined experimentally exhibit falloff

behavior which is “broader” than that predicted by the Lindemann mechanism. Troe and

co-workers have empirically developed an analytic expression to describe unimolecular

rate coefficients in the falloff regime [200]. In the Troe formulation the first-order rate

coefficient, k1, is given by

k1 = FL F k∞

where FL defines the Lindemann falloff relation, k1/k∞ given above, and F is an empirical

pressure broadening factor fit to experimental data or in some cases a theoretical

calculation. The pressure broadening factor, F, is given by Troe and co-workers as

( )

2

log14.0log

1

loglog

⎥⎦

⎤⎢⎣

⎡+−

++

=

CPNCP

FF

r

r

cent

with N = 0.75 – 1.27logFcent, C = -0.4 - 0.67logFcent and Pr = ke[M]/k∞. The pressure

broadening factor, F, is defined in terms of a temperature-dependent variable Fcent

Fcent = (1 – a) exp(-T / b) + a exp(-T / c) + exp(-d / T)

Fcent is a factor (determined from fit to experimental data) by which the rate coefficient of

a unimolecular reaction at temperature T and reduced pressure of Pr = ke[M]/k∞ = 1.0 is

less than the value k∞/2 which it would have if it behaved according to the Lindemann

formulation. CHEMKIN [135] allows the user to input the parameters a, b, c, and d for a

given reaction to prescribe Fcent and the pressure broadening factor, F, is calculated via

the above Troe formulation.

42

n(E

)

Energy

Figure 3.1: Reactant population energy distribution under high-pressure-limit conditions (solid) and under falloff conditions (dashed). Under falloff conditions the unimolecular dissociation step occurs at a rate fast enough such that the equilibrium Boltzmann distribution cannot be maintained by collisions and the population is depleted above the reaction threshold.

43

Chapter 4: Alkane decomposition

This chapter describes the measurement of rate coefficients for the decomposition

of ethane, propane, iso-butane, and n-butane, reactions (1)-(5). These measurements were

made in the shock tube facility using the CH3 laser absorption diagnostic at 216.62 nm

described in Chapter 2.

4.1 Determination of rate coefficients Experiments were performed behind incident and reflected shock waves at

temperatures of 1297 to 2034 K and pressures of 0.13 to 8.8 atm for the measurement of

k1-k5 with mixture concentrations ranging from 100 to 400 ppm of the four alkanes of

interest (ethane, propane, iso-butane, and n-butane) dilute in argon. Narrow-linewidth

laser absorption was used for ~ppm sensitive CH3 detection. The low concentrations

provided isolated sensitivity to the decomposition rate coefficients of interest.

Additionally, rate coefficients were only fit to the early time behavior of the CH3 profiles

to maintain kinetic isolation of the decomposition reactions and to avoid interfering

absorption from other product species that are formed at longer times.

In order to determine rate coefficients for reactions (1)-(5), the measured CH3

mole fractions were fit using detailed kinetic models. Fortunately, the low initial

concentrations used in these experiments allowed fits that are fairly insensitive to the

choice of rate coefficients for reactions other than reactions (1)-(5). In the case of ethane

and propane the GRI Mech 3.0 [95] mechanism was used, for iso-butane the mechanism

of Wang et al. [133] was used, and for n-butane the mechanism of Marinov et al. [134]

was used. The only rate coefficients changed in these mechanisms, for the purpose of

fitting the experimental data, are those of reactions (1)-(5). Computations were done

using CHEMKIN 2.0 and SENKIN [135-136].

In the case of ethane and propane the kinetic modeling is quite simple, the initial

rise of CH3 provides a direct fit to reaction (1) or (2), as there is only one decomposition

45

path for these molecules at the conditions of these experiments. Additionally, during the

early rise, interfering absorption due to C2H4 and C2H2 is negligible; the absorption

coefficients for these two species have been measured by several groups [37,98,137] and

were verified with good agreement prior to the current experiments [118]. An example

data trace and sensitivity plot for ethane decomposition is shown in Figure 4.1. Notice

that interfering reactions have essentially no influence on the fit of reaction (1). An

example propane decomposition experiment and sensitivity is shown in Figure 4.2. Here

the primary interfering reaction is C3H8 + H and, because of the low initial propane

concentration, it has little effect on the fit of reaction (2).

For the iso-butane decomposition experiments the initial rise of CH3 provides a

direct fit to reaction (3). The rate coefficient was fit only to the early time behavior of the

CH3 traces because at longer times a complication arises from interfering absorption.

Example data and sensitivity are given in Figure 4.3; notice that the model fits the first 75

µs of the trace well but then diverges slightly from the experiment at longer times due to

interfering absorption. Experiments were performed to identify possible interfering

absorbing species (C3H6, C2H4, and C2H2) in iso-butane decomposition [118]. The

absorption coefficients of these species at 216.62 nm were measured by monitoring

216.62 nm laser absorption during shock heating. It was found that although none of

these species had absorption coefficients large enough to be of concern at the early times

where the iso-butane decomposition rate coefficient was determined, the decomposition

products of C3H6 showed moderately strong absorption. These C3 decomposition

products are likely responsible for the interference in the iso-butane decomposition

experiments; similar interference was reported by Koike and Morinaga [36].

Analysis of the CH3 profiles obtained from the n-butane experiments is

complicated by the fact that there are two parallel channels for n-butane decomposition,

reactions (4) and (5), and thus the CH3 profiles must be fit in terms of the overall

decomposition rate coefficient, k4+k5, and the branching ratio, k4/(k4+k5). The overall rate

of decomposition, k4+k5, is determined by fitting to the early-time behavior of the CH3

concentration profile. At longer times the CH3 concentration rolls off to a peak before

methyl-methyl recombination brings about a decay in the CH3 concentration. The peak in

the methyl concentration is controlled by the branching ratio, k4/(k4+k5), and thus the

46

branching ratio can be inferred by fitting to the peak in CH3 concentration. See Figure 4.4

for an example data trace and sensitivity plot. Interfering chemistry at longer times

provides a bit more uncertainty in the fits (sensitivity in Figure 4.4), but fortunately the

primary interfering reaction is methyl recombination which has been measured here in

reverse and is relatively well-known (±20%). Interfering absorption is negligible in the n-

butane decomposition case because the subsequent C3 product fragments are in lower

concentration than in the iso-butane case.

The present rate coefficient determinations for reactions (1)-(5) are tabulated in

Appendix A and shown in Figures 4.5-4.9 on Arrhenius plots along with least-squares

fits; the experimental scatter about the least-squares fits is small (±10% in k1, ±7% in k2,

±7% in k3, ±12% in k4, and ±21% in k5). The primary contributions to uncertainties in the

rate coefficients are: temperature, CH3 absorption coefficient, fitting the data to computed

profiles, and uncertainties resulting from interfering chemistry. These uncertainties give

overall uncertainties in k1-k5 of ±19%, ±26%, ±25%, ±31%, and ±31%, respectively. See

Appendix B for details of the uncertainty analysis.

4.2 RRKM / master equation calculations Single-channel (ethane, propane, and iso-butane) and multi-channel (n-butane)

RRKM calculations with numerical solution to the 1-D (internal energy) master equation

were carried out using a restricted (hindered) Gorin model for the transition state for the

decomposition of the four alkanes of interest. These calculations provided falloff curves

that were fit to the current alkane decomposition rate coefficient determinations and in

the case of ethane and propane the calculations were also fit to two previous

recombination studies. The methyl/methyl recombination data of Slagle et al. [23] (296-

906 K) and the methyl/ethyl recombination data of Knyazev and Slagle [31] (301-800 K)

were fit respectively with RRKM / master equation calculations to extend the findings for

ethane and propane decomposition to lower temperatures.

Two parameters were adjusted in the calculations to provide a best fit to the

experimental rate coefficient determinations: the hindrance, η, on the restricted internal

rotations in the transition state and the energy transfer parameter, α, in the exponential

down energy transfer model. Previous studies [16,138-139] have reported that the

47

hindrance parameter follows the functional form, η = a + bT -1/6. This functional form was

found to hold for both iso-butane and n-butane decomposition over the temperature range

of the experiments performed in these studies (1310-1551 K for iso-butane and 1297-

1601 K for n-butane). However, the large temperature-range of data fit for ethane (296-

1924 K) and propane (301-1653 K) decomposition could not be fit using the above

simple functional form of the hindrance parameter. For the broad temperature range

examined for ethane and propane the hindrance parameter was found to asymptote from

high-temperature to a nearly constant value at lower temperatures; see Figure 4.5.

Lennard-Jones parameters for the various species were taken from [140]; the molecular

frequencies [141-143], moments of inertia [16,33,143-145], heats of formation [146-147],

and barrier heights [144,147-150] were taken from various references for the RRKM /

master equation calculations and are listed in Table 1.

The exponential-down collisional energy transfer model was used with a

temperature-dependent collisional energy transfer parameter, α, which was fit to the

current experiments and low-temperature recombination data. The resulting best-fit

expressions for the energy collisional transfer parameter for argon collisions with the four

alkanes are also given in Table 1. The falloff behavior of the current experiments could

not be accurately fit without a temperature-dependent α; similar results have been given

by Klippenstein and Harding [22], Knyazev and Slagle [31,151] and Knyazev and Tsang

[152].

4.3 Analytic Arrhenius / Troe expressions for rate

coefficients The RRKM/master equation calculations provided rate coefficients in both the

high-pressure and low-pressure limits for reactions (1)-(5) which were fit using two- and

three-parameter analytic Arrhenius expressions and the Troe formulation for the pressure

broadening falloff factor, Fcent. Accurate three parameter analytic expressions (ATnexp(-

E/T)) for k∞ and k0 over large temperature ranges are difficult to develop due in part to the

simplicity of the pre-exponential temperature dependence term, which is a function of the

change in specific heat from reactant to transition state <∆C‡p>. One coefficient to

48

represent the heat capacity term over a large range of temperatures is simply not enough

(most polynomial expressions for heat capacitance contain five terms). Therefore, in the

case of ethane and propane analytic expressions for reactions (1) and (2), in the Troe

formulation, are given over a smaller temperature range than that for which the RRKM

calculations were carried out and the resulting pre-exponential temperature dependences

are large. The reaction (1) rate expressions for 700 to 1924 K are:

k∞,1(T) = 1.88 x 1050 T -9.72 exp(-54020 K/T) [s-1]

k0,1(T) = 3.72 x 1065 T -13.14 exp(-51120 K/T) [cm3mol-1s-1]

Fcent,1(T) = 0.61 exp(-T/100 K) + 0.39 exp(-T/1900 K) + exp(-6000 K/T)

The reaction (2) rate expressions for 600 to 1653 K are:

k∞,2(T) = 1.29 x 1037 T -5.84 exp(-49010 K/T) [s-1]



Analytic expressions for the decomposition of iso-butane and n-butane, reactions (3)-(5),

are given over the smaller temperature range of the current experimental findings and

therefore, are given in the two-parameter Arrhenius form. The reaction (3) rate

expressions for 1320 to 1560 K are:

k∞,3(T) = 4.83 x 1016 exp(-40210 K/T) [s-1]

k0,3(T) = 2.41 x 1019 exp(-26460 K/T) [cm3mol-1s-1]

Fcent,3(T) = 0.75 exp(-T/750 K)


k∞,4(T) = 4.28 x 1014 exp(-35180 K/T) [s-1]

k0,4(T) = 5.34 x 1017 exp(-21620 K/T) [cm3mol-1s-1]

Fcent,4(T) = 0.28 exp(-T/1500 K)

And the reaction (5) rate expressions for 1320 to 1600 K are:

k∞,5(T) = 2.72 x 1015 exp(-38050 K/T) [s-1]

k0,5(T) = 4.72 x 1018 exp(-24950 K/T) [cm3mol-1s-1]

Fcent,5(T) = 0.28 exp(-T/1500 K)

It should be noted that these analytic expressions, fit to the RRKM / master equation

calculations, differ from those calculations by no more than ±10%. Figures 4.11-4.15

show the current data, RRKM calculations, and selected previous experimental studies on

49

falloff plots; Figures 4.16-4.20 show the current high-pressure-limit rate coefficients with

comparison to selected theoretical studies.

4.4 Discussion 4.4.1 Comparison with previous experimental studies The current findings for reactions (1)-(4) are compared with previous

experimental studies in Figures 4.11-14; there are no previous high-temperature

experimental results for comparison to the reaction (5) results shown in Figure 4.15. The

current findings for reaction (1) are in good agreement with the dissociation data of

Davidson et al. [11], although the current experimental scatter is much improved. The

low pressure (~1 atm) recombination data of Du et al. [13] and the high pressure (~50-

100 atm) recombination data of Hwang et al. [12] show fair agreement with the current

study, although their results are both slightly lower than the current data and RRKM fits.

The current findings for reaction (2) are in good agreement with the work of Al-

Alami and Kiefer [27]. At low-temperatures the current findings are in good agreement

with the work of Simmie et al. [29] but at higher temperatures their results are

significantly lower and their scatter increased. Both the measurements of Hidaka et al.

[30] and Chiang and Skinner [26] are significantly lower than the current findings. Of

concern are the high activation energies reported by Hidaka et al. and Chiang and Skinner

for their 2 atm (falloff) experiments. Their data cannot be reconciled with the present

findings or other theoretical results.

The current findings for reaction (3) are in fairly good agreement with the iso-

butane 3.39 µm laser absorption measurements of Hidaka et al. [35], particularly at lower

temperatures. A comparison of the current k3 and k4 results with both studies by Koike

and Morinaga [36-37] shows poor agreement. Koike and Morinaga measured CH3 using

lamp absorption at 216 nm during the shock heating of mixtures of 1-2.8% iso-butane or

n-butane dilute in argon. The large initial concentrations of reactant used in their

experiments provide results that are highly sensitive to the choice of the other rate

coefficients in their reaction mechanism. Their low findings for k3 and k4, in comparison

to the current findings, is likely due to their choice of a slow rate for methyl

50

recombination, which is of significant sensitivity in their experiments. Koike and

Morinaga [36-37] employ a rate for methyl recombination that is a factor of two slower at

1450 K than the currently accepted rate, which has an uncertainty of ±20%.

4.4.2 Comparison with previous theoretical calculations As stated above, the current data for reactions (1)-(5) was fit with RRKM/master

equation calculations by adjusting two parameters: the collisional energy transfer

parameter, α (which influences falloff), and the hindrance parameter for the internal

rotors, η, in the transition state (which influences k∞). The current data and RRKM

analysis suggests an increase in α with temperature and a stronger falloff in k∞ (increased

hindrance) at higher temperatures than the previous theoretical studies predict for both

reactions (1) and (2), see Figures 4.16-4.17. The previous theoretical models do a good

job of predicting the low-scatter data of Slagle et al. [23] and Knyazev and Slagle [31],

but previous high-temperature experimental data had excessive scatter which made

accurate theoretical fitting difficult; in fact, many theoretical studies predict high-

temperature rates that were simply extrapolated from the low-temperature results.

A comparison of the current high-pressure-limit rate coefficients for reactions (3)-

(5) can be found in Figures 4.18-4.20. The current findings for k∞,3 are about a factor of 2

to 3 greater than the theoretical RRKM prediction of Tsang [17] and the review

recommendation of Warnatz [153] but show essentially the same activation energy. The

RRKM expression given by Tsang [17] was guided by the only high-temperature data

available at the time that of Koike and Morinaga [36], thus Tsang’s high-pressure-limit

recommendation is lower than the current finding.

The current findings for k∞,4 are in good agreement with the QRRK predictions of

Dean [14] and the review recommendations of Warnatz [153] although the activation

energy of the current study is somewhat lower. The lower activation energy for n-butane

at high-temperatures is consistent with the results for ethane and propane decomposition

where the temperature-range of data is more complete. At high-temperatures the

increased hindrance on the transition state internal rotations causes a reduction in the

high-pressure-limit rate coefficient and the high-pressure-limit rate coefficient falls off

51

more at high-temperature than a simple extrapolation from low-temperature would

predict.

The current findings for k∞,5 are about a factor of three smaller than the QRRK

predictions of Dean [14] but in good agreement with the review recommendations of

Warnatz [153]. The results of the current study for reactions (4) and (5) give a branching

ratio, k4/(k4+k5), that is approximately 0.5 (1320-1600 K), in agreement with the

recommendation of Warnatz [153].

In addition the results of the current study for ethane and propane decomposition,

k1 and k2, are compared with the most recent International Union of Pure and Applied

Chemistry (IUPAC) recommendations (Baulch et al. [201]) in Figures 4.11, 4.12, 4.16,

and 4.17. The IUPAC recommendation for ethane decomposition is in relatively good

agreement with the current study. However, the IUPAC recommendation for propane

decomposition is in large disagreement in the falloff regime, Figure 4.12. The IUPAC

recommendation suggests a much more dramatic falloff for propane, with rate

coefficients that are significantly lower than the current data in the falloff regime.

4.4.3 Geometric mean rule Additionally, the measurements of k1, k2, and k5 allow a test of the validity of the

geometric mean rule [31,154-155]. The geometric mean rule is often used to predict rate

coefficients for decomposition or radical cross reactions, when data are not available and

is written

kAB = 2(kAAkBB)1/2

where kAB is the rate coefficient of A + B, kAA is the rate coefficient of A + A, and kBB is

the rate coefficient of B + B. This equation can easily be arranged for the rate coefficient

for n-C4H10 → C2H5 + C2H5 in terms of the rate coefficients for C2H6 → CH3 + CH3 and

C3H8 → CH3 + C2H5 and the appropriate equilibrium constants. The resulting rate

coefficient for reaction (5) predicted by the geometric mean rule is in good agreement

with the current measurements, see Figure 4.20. This test, although limited, suggests that

the geometric mean rule is a good estimator for high-temperature alkane decomposition

rates.

52

Table 4.1: Parameters used in RRKM / master equation calculations for alkane decomposition.

C2H6 Heat of Formation (0K) [146]: -16.34 kcal/mol Moments of Inertia (amu Å2) [16]: 25.3 (inactive); 25.3 (inactive); 6.3 (active), σ = 3 Frequencies (cm-1) [141]: 2977(4), 2954, 2896, 1469(4), 1382(2), 1190(2), 995,

822(2), 260 CH3 … CH3

Frequencies (cm-1) [141]: 3184(4), 3002(2), 1384(4), 580(2) Active 1-D external rotor (amu Å2) [16]: 6.3, σ = 3 Active 1-D internal torsion (amu Å2) [145]: 1.777, σ = 3Active 2-D internal rotor (amu Å2) [145]: 1.760, σ = 2; η (see Fig. 5) Active 2-D internal rotor (amu Å2) [145]: 1.760, σ = 2; η (see Fig. 5)Barrier at 0 K [146]: 87.6 kcal/mol Energy transfer parameter: α = 0.8 x (T [K]) cm-1

C3H8Heat of Formation (0K) [148]: -19.69 kcal/mol Moments of Inertia (amu Å2) [144]: 67.75 (inactive); 59.84 (inactive); 17.3 (active), σ = 1 Frequencies (cm-1) [141]: 2977, 2973, 2968(2), 2967, 2962, 2887(2), 1476,

1472, 1464, 1462, 1451, 1392, 1378, 1338, 1278, 1192, 1158, 1054, 940, 922, 869, 748, 369, 268, 216

CH3 … C2H5Frequencies (cm-1) [141-142]: 3184(2), 3002, 1384(2), 580, 3112, 3033, 2987(2), 2942, 1440(3), 1366, 1175(2), 1138, 528(3) Active 1-D external rotor (amu Å2) [144]: 17.3, σ = 1 Active 1-D internal torsion (amu Å2) [33,145]: 3.1, σ = 3 Active 2-D internal rotor (amu Å2) [145]: 1.76, σ = 2; η (see Fig. 5)Active 1-D internal rotor (amu Å2) [33]: 4.87, σ = 1; η (see Fig. 5)Active 1-D internal rotor (amu Å2) [33]: 22.3, σ = 1; η (see Fig. 5)Barrier at 0 K [146-147]: 87.9 kcal/mol Energy transfer parameter: α = 1.2 x (T [K]) cm-1

i-C4H10 Heat of Formation (0K)[147]: -25.43 kcal/molMoments of Inertia (amu Å2)[144]: 114.1 (inactive); 67.0 (inactive); 64.9 (active), σ = 1 Frequencies (cm-1)[143]a: 2987, 2921, 2898, 1469, 1388, 1171, 769, 418, 2986,

1439, 922, 204, 2990, 2978, 2914, 1465, 1445, 1361, 1314, 1152, 938, 891, 347, 248, 2990, 2978, 2914, 1465, 1445, 1361, 1314, 1152, 938, 891, 347, 428

CH3 … i-C3H7Frequencies (cm-1)[141,143]a: 3057, 2985, 2926, 2846, 1448, 1437, 1372, 1139,

998, 854, 394, 347, 122, 2986, 2924, 2841, 1436, 1427, 1369, 1324, 1110, 910, 906, 112, 3184, 3184, 3002, 1384, 1384, 580

Active 1-D external rotor (amu Å2)[144]: 64.9, σ = 1 Active 1-D internal torsion (amu Å2)[143,145]b: 3.38, σ = 3Active 2-D internal rotor (amu Å2)[145]: 1.76, σ = 2; η = 265.4 – 616.9 T -1/6

Active 1-D internal rotor (amu Å2)[143]b: 13.49, σ = 1; η = 265.4 – 616.9 T -1/6


Barrier at 0 K[144]: 86.8 kcal/mol Energy transfer parameter: α = 0.8 x (T [K]) cm-1

53

n-C4H10

Heat of Formation (0K)[147]: -23.55 kcal/molMoments of Inertia (amu Å2)[143]b: 149.2 (inactive); 140.2 (inactive); 21.6 (active), σ = 1 Frequencies (cm-1)[141]: 2965, 2872, 2853, 1460, 1442, 1382, 1361, 1151,

1059, 837, 425, 2968, 2930, 1461, 1257, 948, 731, 194, 102, 2965, 2912, 1460, 1300, 1180, 803, 225, 2968, 2870, 2853, 1461, 1461, 1379, 1290, 1009, 964, 271

CH3 … n-C3H7Frequencies (cm-1)[141,143]a: 3033, 2994, 2924, 2908, 1463, 1442, 1425, 1359,

1283, 1058, 991, 864, 506, 323, 3131, 3002, 2945, 1453, 1264, 1156, 862, 720, 246, 67, 3184, 3184, 3002, 1384, 1384, 580

Active 1-D external rotor (amu Å2)[143]b: 21.56, σ = 1 Active 1-D internal torsion (amu Å2)[143,145]b: 3.37, σ = 3Active 2-D internal rotor (amu Å2)[145]: 1.76, σ = 2; η = 439.5 – 1186.5 T -1/6



Barrier at 0 K[147,149-150]: 88.5 kcal/mol C2H5 … C2H5

Frequencies (cm-1)[142]: 3112, 3033, 2987, 2987, 2984, 1440, 1440, 1440, 1366, 1175, 1175, 1138, 528, 528, 528, 3112, 3033, 2987, 2987, 2984, 1440, 1440, 1440, 1366, 1175, 1175, 1138, 528, 528, 528

Active 1-D external rotor (amu Å2)[143]b: 21.6, σ = 1 Active 1-D internal torsion (amu Å2)[33,143]b: 12.05, σ = 2Active 1-D internal rotor (amu Å2)[33]: 4.87, σ = 1; η = 282.6 – 672.8 T -1/6

Active 1-D internal rotor (amu Å2)[33]: 22.3, σ = 1; η = 282.6 – 672.8 T -1/6



Barrier at 0 K [144,147]: 87.3 kcal/mol Energy transfer parameter: α = 1.2 x (T [K]) cm-1

a Frequencies for i-C4H10, i-C3H7, and n-C3H7 are from ab initio calculations at the DFT B3LYP/6-31G** level scaled by 0.962. b Moments for n-C4H10, i-C3H7 and n-C3H7 are from ab initio calculations at the DFT B3LYP/6-31G** level.

54

0 10 20 30 40 50 60 70 80

0.0

0.5

1.0

Loca

l CH

3 Sen

sitiv

ity, S

Time [µs]

(1) C2H6<=>2CH3a CH3+C2H6<=>C2H5+CH4b 2CH3<=>H+C2H5

(1)

ab

0

25

50

75

100

C

H3 M

ole

Frac

tion

[ppm

]

Figure 4.1: Example ethane data, modeling, and sensitivity. Reflected shock conditions: 1589 K, 1.664 atm, 202 ppm C2H6. Top graph: solid line, fit to data using GRI Mech 3.0 [95] and adjusting k1; dashed lines, variation of k1 ±50%. The spike at 0 µs is caused by deflection of the diagnostic beam by the reflected shock wave. S = (dXCH3 / dki) (ki / XCH3,local), where ki is the rate constant for reaction i and XCH3,local is the local CH3 mole fraction.

0 10 20 30 40 50 60

0.0

0.5

1.0

Loca

l CH

3 Sen

sitiv

ity, S

Time [µs]

(2) C3H8<=>CH3+C2H5a H+C3H8<=>C3H7+H2b H+CH3(+M)<=>CH4(+M)c C2H6<=>2CH3

(2)

a

bc

0

10

20

30

40

50

CH

3 Mol

e Fr

actio

n [p

pm]

Figure 4.2: Example propane data, modeling, and sensitivity. Reflected shock conditions: 1433 K, 4.535 atm, 201 ppm C3H8. Top graph: solid line, fit to data using GRI Mech 3.0 [95] and adjusting k2; dashed lines, variation of k2 ±50%; dotted lines, variation of rate coefficient for H + C3H8 → C3H7 + H2 by ±50%. S = (dXCH3 / dki) (ki / XCH3,local), where ki is the rate constant for reaction i and XCH3,local is the local CH3 mole fraction.

55

0 25 50 75 100 125 150

0.0

0.5

1.0

Loca

l CH

3 Sen

sitiv

ity, S

Time [µs]

(3) IC4H10<=>CH3+IC3H7a IC4H10+H<=>IC4H9+H2b C3H6+H<=>C2H4+CH3c CH2(S)+H2<=>CH3+Hd IC4H10+H<=>TC4H9+H2e 2CH3(+M)<=>C2H6(+M)

(3)

ab

c d e

0

50

100

150

200

250

CH

3 Mol

e Fr

actio

n [p

pm]

Figure 4.3: Example iso-butane data, modeling (top), and sensitivity (bottom). Incident shock conditions: 1401 K, 0.238 atm, 400 ppm i-C4H10/Ar. Top graph: solid line, fit to data using Wang et al. [133] mechanism and adjusting k3; dashed lines, variation of k3 ±50%; dotted lines, variation of rate coefficient for i-C4H10 + H → i-C4H9 + H2 by ±50%. S = (dXCH3 / dki) (ki / XCH3,local), where ki is the rate constant for reaction i and XCH3,local is the local CH3 mole fraction.

0 10 20 30 40 50 60 70 80

0.0

0.5

1.0

Loca

l CH

3 Sen

sitiv

ity, S

Time [µs]

(4)+(5) nC4H10 => productsBR branching ratio k4/(k4+k5)b C4H10+H<=>SC4H9+H2c 2CH3(+M)<=>C2H6(+M)

(4)+(5)

BR

b

c

0

25

50

75

100

125

150

CH

3 Mol

e Fr

actio

n [p

pm]

Figure 4.4: Example n-butane data, modeling (top), and sensitivity (bottom). Reflected shock conditions: 1479 K, 1.673 atm, 200 ppm n-C4H10/Ar. Top graph: solid line, fit to data using Marinov et al. [134] mechanism and adjusting overall decomposition rate, k4+k5, and branching ratio, k4/(k4+k5); dashed lines, variation of k4+k5 ±50%; dotted lines, variation of k4/(k4+k5) ±25%. S = (dXCH3 / dki) (ki / XCH3,local), where ki is the rate constant for reaction i and XCH3,local is the local CH3 mole fraction.

56

0.50 0.55 0.60 0.65 0.70 0.75102

103

104

105

0.22 atm

k 1 [1/s

]

1000/T [1/K]

0.13 atm

Figure 4.5: Experimental data and least-squares fits for C2H6 → CH3 + CH3. Filled squares, 1.6 atm data, k1 = 1.08x1012 exp(-31200K/T); filled triangles, 4.2 atm data, k1 = 4.01x1012 exp(-32400K/T); filled stars, 7.5 atm data, k1 = 1.55x1013 exp(-34200K/T); open squares, 0.13-0.22 atm data.

0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74102

103

104

105

0.23 atm

k 2 [1/s

]

1000/T [1/K]

0.19 atm

Figure 4.6: Experimental data and least-squares fits for C3H8 → CH3 + C2H5. Filled squares, 1.7 atm data, k2 = 1.97x1013 exp(-32400K/T); filled triangles, 4.3 atm data, k2 = 5.10x1013 exp(-33400K/T); filled stars, 7.9 atm data, k2 = 4.97x1013 exp(-33100K/T); open squares, 0.19-0.23 atm data.

57

0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80102

103

104

105

106

k 3 [1/s

]

1000/T [1/K]

0.21 atm

0.25 atm

Figure 4.7: Experimental data and least-squares fits for i-C4H10 → CH3 + i-C3H7. Filled squares, 1.7 atm data, k3 = 4.98x1014 exp(-34800K/T) s-1; filled triangles, 4.4 atm data, k3 = 1.71x1015 exp(-36300K/T) s-1; filled stars, 8.4 atm data, k3 = 3.59x1015 exp(-37000K/T) s-1; open squares, 0.21-0.25 atm data.

0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80102

103

104

105

106

k 4 [1/s

]

1000/T [1/K]

0.20 atm

0.25 atm

Figure 4.8: Experimental data and least-squares fits for n-C4H10 → CH3 + C3H7. Filled squares, 1.7 atm data, k4 = 4.15x1013 exp(-32800K/T) s-1; filled triangles, 4.2 atm data, k4 = 4.59x1014 exp(-35700K/T) s-1; filled stars, 8.3 atm data, k4 = 3.33x1014 exp(-35300K/T) s-1; open squares, 0.20-0.25 atm data.

58

0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80102

103

104

105

106

k 5 [1/s

]

1000/T [1/K]

0.20 atm

0.25 atm

Figure 4.9: Experimental data and least-squares fits for n-C4H10 → C2H5 + C2H5. Filled squares, 1.7 atm data, k5 = 3.82x1014 exp(-36200K/T) s-1; filled triangles, 4.2 atm data, k5 = 9.30x1015 exp(-40200K/T) s-1; filled stars, 8.3 atm data, k5 = 5.21x1015 exp(-39400K/T) s-1; open squares, 0.20-0.25 atm data.

200 400 600 800 1000 1200 1400 1600 1800 200050

60

70

80

90

100

η, h

indr

ance

par

amet

er [%

]

T [K]

Figure 4.10: Hindrance parameter temperature dependence. Filled squares, RRKM fit to k1 data of current study; filled circles, RRKM fit to k2 data of current study; open squares, RRKM fit to k1 data of Slagle et al. [23]; open circles, RRKM fit to k2 data of Knyazev and Slagle [31].

59

0.01 0.1 1 10 100 1000100

101

102

103

104

105

106

107

Current Study 1425K 1503K 1720K 1924KDavidson et al. 1503K 1720K 1924KDu et al. 1425K 1503K 1720KHwang et al. 1425K 1503K

k 1 [1/s

]

Pressure [atm]

1425 K

1503 K

1720 K

1924 K

Figure 4.11: Falloff plot for C2H6 → CH3 + CH3. Solid symbols, current study; open symbols, Davidson et al. [11]; open symbols with x, Du et al. [13]; half filled symbols, Hwang et al. [12]; solid lines, RRKM/master equation results with α = 0.8 x (T [K]) cm-1; dashed lines, Baulch et al. [201] IUPAC recommendation.

0.01 0.1 1 10 100 1000100

101

102

103

104

105

106

Current Study 1365K 1435K 1572K 1653KAl-Alami and Kiefer 1435K 1572K 1653KSimmie et al. 1365K 1435K 1572K 1653KHidaka et al. 1365K 1435KChiang and Skinner 1365K 1435K

k 2 [1/

s]

Pressure [atm]

1653 K

1572 K

1435 K

1365 K

Figure 4.12: Falloff plot for C3H8 → CH3 + C2H5. Solid symbols, current study; open symbols, Al-Alami and Kiefer [27]; half filled symbols, Simmie et al. [29]; open symbols with line, Hidaka et al. [30]; open symbols with x, Chiang and Skinner [26]; solid lines, RRKM/master equation results with α = 1.2 x (T [K]) cm-1; dashed lines, Baulch et al. [201] IUPAC recommendation.

60

1E-3 0.01 0.1 1 10 100 1000102

103

104

105

106

Current Study 1539K 1458K 1401K 1352KHidaka et al. 1539K 1458K 1401K 1352KKoike and Morinaga 1539K 1458K 1401K 1352K

k 3 [1/s

]

Pressure [atm]

Figure 4.13: Falloff plot for i-C4H10 → CH3 + i-C3H7. Solid symbols, current study; open symbols, Hidaka et al. [35]; half filled symbols, Koike and Morinaga [36]; solid lines, RRKM/master equation results with α = 0.8 x (T [K]) cm-1.

1E-3 0.01 0.1 1 10 100 1000101

102

103

104

105

Current Study 1560K 1500K 1432K 1381K 1337KKoike and Morinaga 1560K 1500K 1432K 1381K 1337K

k 4 [1/s

]

Pressure [atm]

Figure 4.14: Falloff plot for n-C4H10 → CH3 + n-C3H7. Solid symbols, current study; open symbols, Koike and Morinaga [37]; solid lines, RRKM/master equation results with α = 1.2 x (T [K]) cm-1.

61

1E-3 0.01 0.1 1 10 100 1000101

102

103

104

105

Current Study 1560K 1500K 1432K 1381K 1337K

k 5 [1/s

]

Pressure [atm]

Figure 4.15: Falloff plot for n-C4H10 → C2H5 + C2H5. Solid symbols, current study; solid lines, RRKM/master equation results with α = 1.2 x (T [K]) cm-1.

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.410-12

10-10

10-8

10-6

10-4

10-2

100

102

104

106

0.50 0.55 0.60 0.65 0.70 0.75102

103

104

105

106

107

k ∞,1 [1

/s]

1000/T [1/K]

Figure 4.16: High pressure rate coefficient for C2H6 → CH3 + CH3 (700-1924K). Symbols, current RRKM results (listed on figure); solid line, fit to current RRKM results; dashed line, Klippenstein and Harding [22]; dotted line, Hessler and Ogren [20]; dot-dashed line, Wagner and Wardlaw [15]; gray line, Baulch et al. [201] IUPAC recommendation.

62

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.710-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105

107

0.60 0.65 0.70 0.75103

104

105

106

k ∞,2 [1

/s]

1000/T [1/K]

Figure 4.17: High pressure rate coefficient for C3H8 → CH3 + C2H5 (600-1653 K). Symbols, current RRKM results (listed on figure); solid line, fit to current RRKM results; dashed line, Mousavipour and Homayoon [33]; dotted line, Knyazev and Slagle [31]; dot-dashed line, Tsang [17]; dot-dot-dashed, Dean [14] ; gray line, Baulch et al. [201] IUPAC recommendation.

0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80103

104

105

106

k ∞,3 [1

/s]

1000/T [1/K]

Figure 4.18: High-pressure rate coefficient for i-C4H10 → CH3 + i-C3H7. Symbols, current RRKM results (listed on figure); solid line, fit to current RRKM results; dashed line, Tsang [17]; dotted line, Warnatz [153].

63

0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80102

103

104

105

106

k ∞,4 [1

/s]

1000/T [1/K]

Figure 4.19: High-pressure rate coefficient for n-C4H10 → CH3 + n-C3H7. Filled squares, current RRKM results (listed on figure); solid line, fit to current RRKM results; dashed line, Dean [14]; dotted line, Warnatz [153].

0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80102

103

104

105

106

k ∞,5 [1

/s]

1000/T [1/K]

Figure 4.20: High-pressure rate coefficient for n-C4H10 → C2H5 + C2H5. Filled circles, current RRKM results (listed on figure); solid line, fit to current RRKM results; dashed-dot line, Dean [14]; dashed-dot-dot line, Warnatz [153]; dashed line, result using the geometric mean rule.

64

Chapter 5: CO2 incubation and

decomposition

This chapter discusses the measurement of incubation times and the rate

coefficient for CO2 decomposition, reaction (6). These measurements were made in the

shock tube facility using CO2 laser absorption at 216.5 and 244 nm described in Chapter

2.

5.1 Determination of incubation times and

decomposition rate coefficients Mixtures of 1% and 2% CO2 dilute in argon were heated behind reflected shock

waves in the shock tube facility described in Chapter 2. The incident shock conditions

ranged from 1547 to 2123 K and 0.08 to 0.2 atm and the reflected shock conditions

ranged from 3209 to 4601 K and 0.44 to 0.98 atm. The reflected shock pressure and

temperature were taken to be the vibrationally equilibrated conditions and have

uncertainties of ~1% and ~0.7% respectively. Despite the observation of incubation

times, vibrational relaxation in the lower energy levels occurs rapidly behind the incident

and reflected shock waves [109-110]. The vibrational relaxation times behind the incident

and reflective waves for the conditions of these experiments are 9 to 11 µs and 0.7 to 0.9

µs respectively, based on the vibrational relaxation measurements of Simpson et al.

[109].

The progress of reaction was monitored by observing laser absorption by CO2 at

216.5 and 244 nm using the previously measured absorption cross-sections for CO2 at

these two wavelengths [156] as described in Chapter 2. The absorbance for an example

experiment at 216.5 nm is shown in Figure 5.1. The passage of the incident shock wave

causes a schlieren spike in the signal, due to steering of the beam off the detector, after

65

which the CO2/Ar mixture vibrationally relaxes to the vibrationally equilibrated

temperature. The vibrational relaxation causes an induction in the CO2 absorption.

However, the vibrational relaxation after the passage of the incident shock wave was

always sufficiently fast (τrel = 9-11 µs) for the use of the vibrationally equilibrated

incident shock temperature in the calculation of the reflected shock conditions.

Additionally, the observed induction times in the absorbance after the incident shock

waves are in good agreement (~10%) with previous measurements of vibrational

relaxation times by Simpson et al. [109] and Kamimoto and Matsui [110]. After the

passage of the reflected shock wave the absorbance quickly rises due to the increase in

temperature and pressure and the vibrational relaxation is not resolved. Based on the

previous measurement by Simpson et al. [109], the vibrational relaxation time behind the

reflected shock wave in this example (Figure 5.1) is 0.7 µs. While the experimental time

resolution is not sufficient to observe the vibrational relaxation behind the reflected shock

wave a pronounced incubation period is observed prior to the onset of steady

dissociation.

The bond energy of CO2 is quite large (125.7 kcal/mol, 43970 cm-1) in

comparison to the average energy, therefore only a small fraction of the energy

distribution plays a role in the dissociation. Dissociation can only proceed when

vibrational levels are populated near the threshold, and a steady dissociation rate is not

established until the shock-heated CO2 is near a steady-state vibrational distribution.

Establishing population in the highest levels of the vibrational distribution near the

threshold takes considerably more time than is required for vibrational relaxation in the

low-lying levels probed in vibrational relaxation experiments; therefore, the incubation

time is significantly longer than the vibrational relaxation time.

The incubation times were determined from the experimental traces by

extrapolating back the decay in absorbance, due to dissociation, to the steady absorbance

level achieved after the passage of the reflected shock wave (Figure 5.1). With the end of

the incubation period defined as time zero for chemical reaction, the decay in absorbance

was fit by adjusting the CO2 decomposition rate, reaction (6), in the mechanism given in

Table 5.1. Because of the high CO2 concentrations (1-2%) the endothermic

decomposition caused a non-negligible decrease in temperature as the reaction

66

proceeded. The temperature-dependence of the rate coefficient and the absorption cross-

section was taken into account by modeling the decomposition with a constant volume

constraint. The temperature-dependence of the absorption cross-section is known from

previous measurements (Chapter 2 and [156]). However, to treat the temperature-

dependence (activation energy) of the rate coefficient, the activation energy was initially

assumed and the A-factor was adjusted, in an Arrhenius expression, for a best fit to each

experimental absorbance trace. The resulting rate coefficient determinations were fit with

a new Arrhenius expression and the A-factor was again adjusted to provide a best fit to

each experimental trace. This procedure was performed iteratively until the rate

coefficient converged (three iterations). To avoid significant influence on the rate

coefficient determinations by the temperature time-history and the fitting procedure, the

absorption traces were fit over the earliest possible time periods. In fact, if the

temperature history is ignored and the reflected shock conditions are assumed constant

throughout the experiment, the resulting rate coefficients differ only by <15% from those

obtained by accounting for this effect. The rate coefficient results for reaction (6) from

3200 to 4600 K and around 0.5 to 1 atm can be expressed using a two-parameter

Arrhenius representation

k6(T) = 3.14 x 1014 exp(-51300 K / T) [cm3mol-1s-1]

with 1-σ standard deviation of ±8% in the scatter of the data about the fit. The estimated

uncertainty in the rate coefficient determinations is ±21%, see Appendix B. This

uncertainty is primarily based on uncertainty in the temperature history, as the absorption

cross-section has very small uncertainty (<5%), the experimental signal-to-noise is large,

and the two secondary reactions in Table 5.1 have an almost negligible impact on the fits.

The experimental findings for both the second order rate coefficient and incubation times

are shown in Figures 5.2 and 5.3 and a summary of the experimental results is given in

Table A.5.

5.2 Master equation calculations One-dimensional energy-grained master equation calculations, as described in

Chapter 3, were carried out to determine a suitable energy transfer model for description

of both the measured decomposition rate coefficient and the incubation times. The master

67

equation calculations were performed assuming that the translational and rotational CO2

temperatures relax immediately following the passage of the reflected shock wave and

the vibrational temperature is initially at the incident shock value. The vibrational

distribution then evolves due to CO2-Ar collisional energy transfer and the vibrational

states reach a steady distribution before steady dissociation. The computed incubation

times and decomposition rate coefficients were derived in the same manner as they were

from the experiments, by tracking the time-dependent fractional population of CO2.

Due to the sparse density of states of CO2 the master equation calculations were

carried out for a fairly large dimension of 2000 levels over the energy range of 80000

cm-1. A double array was implemented with a grain size of 25 cm-1 for the first 1000

levels and the remaining 1000 levels equidistantly spaced over the entire energy domain

(0-80000 cm-1). Reducing the dimension of the grained master equation significantly

reduced computational time but dramatically influenced the temporal evolution of the

population. However, increasing the level dimension above 2000 resulted in no change to

the computed results.

5.2.1 Energy-specific rate coefficient k(E) The decomposition of CO2 at the conditions of these experiments is nearly in the

low-pressure-limit [45]. Therefore, the decomposition rate coefficient calculated via the

master equation is significantly more sensitive to the collisional energy probability

density function, Pij, than the energy specific rate constant, k(E). Due of the lack of

sensitivity of the rate coefficient and the lack of an accurate potential energy surface the

inverse Laplace transform method was used, described in Chapter 3, to calculate k(E).

The Arrhenius parameters calculated by Yau and Pritchard [157] were used, A∞ = 1.80 x

1012 s-1 and E∞ = 127.3 kcal/mol, for calculation of k(E) via the inverse Laplace transform

method. Reasonable perturbation of these parameters did not substantially affect the

master equation results for the decomposition rate coefficient and incubation times. The

density of states, ρ(E), was calculated using the DenSum code of Barker [123] with the

graining technique described above and the known CO2 frequencies. Additionally, the

CO2 two-dimensional external rotor was considered active in the density of states

68

calculations because of the Coriolis interaction between it and the two degenerate

bending modes [57].

5.2.2 Energy transfer The observed incubation times and rate coefficients are most sensitive to the

form of energy transfer probability density function, Pij, chosen for solution of the master

equation. The exponential model with an energy dependent step size down (Chapter 3)

was chosen:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

)(exp1

j

ij

jij E

EEN

Pα

The parameter α(Ej) was fit to the experimental incubation times and decomposition rate

coefficients with guidance from previous energy transfer studies on triatomics. Although

there is a wealth of literature on CO2 V-T state-to-state transfer at low energies, there

have been no direct studies for CO2 energy transfer at the high energies encountered in

shock tube experiments and important during dissociation. However, there have been

several studies on energy transfer at high energies for other triatomics in rare gases: NO2

[158-159], SO2 [160-161], and CS2 [160,162-163]. All three of these molecules are of

interest because they have enhanced energy transfer at high energies due to the mixing of

excited electronic states with the electronic ground state causing rapid intramolecular

vibrational energy redistribution (IVR) at energies near that of the excited electronic

states. However, enhanced IVR due to mixing of electronic states has not been reported

for CO2. These studies [158-163] also report a dependence on energy that is greater than

linear for the energy transferred per collision, <∆E>, at energies below that where

enhancement due to electronic state mixing occurs. Several of these studies suggest a

quadratic dependence on energy for the energy transferred per collision (<∆E> ~ <E>2)

in the lower energy range [160-163]. It is likely that CO2 behaves similarly to these other

triatomics without the enhancement due to electronic state mixing. Therefore, the form

α(Ej) = C0 + C1Ej2

was chosen for the energy transfer parameter, α, with C0 and C1 as adjustable parameters.

The experimental incubation times and decomposition rate coefficients were best fit by

C0 = 30 cm-1 and C1 = 3.0 x 10-7 (cm-1)-1. Fits were attempted with other models such as

69

Luther and coworker’s empirical generalized exponential model with linear dependence

of α on energy [164]. The observed incubation times were always longer than the

predictions using this model, leading to the same conclusion that Dove and Troe [66]

made in their master-equation study of N2O incubation behind shock waves: the energy

transfer parameter, α, must increase with a greater than linear dependence on energy.

Although the suggested model provides a satisfactory fit of the current experimental

results, the data covers a limited range and is not sufficiently isolated to energy transfer

effects to conclude that this model is complete or unique. For instance, it can not be

determined if the energy transfer parameter, α, varies with temperature. Also, it may be

possible to fit the experimental data with other functional forms of the energy transfer

probability density function, Pij, such as the complete biexponential function. The master

equation results for the incubation times are included in Figure 5.3, an example master

equation calculation of the fractional CO2 concentration is shown in Figure 5.4, and a

summary of the parameters used in the master equation calculations are given in Table

5.2.

5.3 Discussion 5.3.1 CO2 decomposition rate coefficient

A comparison between the current findings for the CO2 second-order

decomposition rate coefficient and several previous experimental studies is shown in

Figure 5.2. The current findings agree very well with the studies of Burmeister and Roth

[41], Ebrahim and Sandeman [43], and Hardy et al. [44]. Although the rate coefficient

result differs in magnitude with Kiefer [46], it does agree on activation energy; however,

the current findings disagree with the activation energies reported by Fujii et al. [42] and

Dean [47]. There has been discussion of the influence of impurities on reported activation

energies for this rate coefficient [44], perhaps this is the reason for the differences in

activation energies of the Fujii et al. [42] and Dean [47] studies. Regardless, the spread in

the rate coefficient at any temperature for all of the previous experiments, in Figure 5.2,

is only a factor of three. If the Kiefer [46] and Dean [47] results are excluded, the spread

is reduced to a factor of less than 1.5. The estimated uncertainties of the current

70

experimental study are small (±21%) and the excellent agreement with references [41],

[43] and [44] gives confidence in the current rate determination.

5.3.2 CO2 incubation Incubation is a collisional phenomenon, therefore, incubation times are plotted in

terms of total Lennard-Jones collisions, ZL-J[M]∆tinc, in Figure 5.3, where ZL-J[M] is the

Lennard-Jones collision rate for CO2-argon collisions (1/s). The number of incubation

collisions was experimentally found to increase with decreasing temperature and can be

expressed as ZL-J[M]∆tinc ~ T-5.1; the temperature-dependence is also satisfactorily

reproduced by the master equation simulations, see Figure 5.3. This temperature-

dependence is very similar to the findings of Hippler and co-workers for methyl radical

incubation [73]; Dove et al. [65] also found strong temperature-dependence in N2O

incubation times. Analytic master equation solutions for incubation in CO2 dissociation have been

reported by Forst and Penner [57] and Yau and Pritchard [58]. In both of these

calculations the exponential energy transfer probability density function, Pij, was used

with a constant energy transfer parameter, α. These studies both predict approximately

2500 incubation collisions at 3500 K for CO2 in a rare gas, an order of magnitude fewer

collisions than was experimentally measured and predicted in the master equation

simulations using the quadratic energy dependence in α. The difference in these master

equation results demonstrates the importance of modeling the energy dependence of α to

predict energy transfer during the activation period. While the present model adequately

explains the observed incubation times and decomposition rate coefficients, experiments

are needed that isolate the energy transferred per collision, <∆E>, at specific energy and

temperature, for improved modeling of chemical reaction systems where energy transfer

plays a dominant role.

71

Table 5.1: Reaction mechanism for CO2 decomposition system. Reaction Rate coefficient [cm3 mol-1 s-1]

and third body efficiencies Reference

(6) CO2 + M → CO + O + M 3.14 x 1014 exp(-51300 K / T) Ar/ 1/ CO2/ 7/ CO/ 3

rate coefficient this study; 3rd body efficiencies from [153]

O2 + CO ↔ O + CO2 2.50 x 1012 exp(-24056 K / T)

[153]

2O + M ↔ O2 + M 1.20 x 1017 T-1.0

Ar/ 0.83/ CO2/ 3.6/ CO/ 1.75 [153]

Table 5.2: Parameters used in master equation calculations for CO2 thermal decomposition. CO2 heat of formation at 0 K -93.96 kcal/mol 2-D moment of inertia (coriolis active) 43.20 amu Å2

CO2 vibrational frequencies (cm-1) 1333, 2349, 667(2) Lennard Jones parameters CO2 σ = 3.47 Å, ε/k = 114 K Lennard Jones parameters Ar σ = 3.94 Å, ε/k = 201 K High pressure limit Arrhenius parameters for inverse Laplace transform calculation of k(E)

A∞ = 1.80 x 1012 s-1, E∞ = 127.3 kcal/mol

Energy transfer parameter α(E) = 30 + 3.0 x 10-7 E2 [cm-1]

-60 -40 -20 0 20 40 60 80 100

0.00

0.05

0.10

0.15

0.20

Abs

orba

nce

Time [µs]

Incubation Time

VibrationalRelaxation

Figure 5.1: Example experiment and fit (dark line): CO2 absorbance (ln(I0/I)) at 216.5 nm. Initial mixture 2% CO2/Ar. Vibrationally equilibrated shock conditions: 1812 K and 0.161 atm. Initial (prior to decomposition) vibrationally equilibrated reflected shock conditions: 3838 K, 0.820 atm.

72

0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34107

108

109

1010

[43]

[44]

[42]

[41]

[46]

present study - data present study - fit Burmeister and Roth [41] Fujii et al. [42] Ebrahim and Sandeman [43] Hardy et al. [44] Kiefer [46] Dean [47]

k [c

m3 m

ol-1 s

-1]

1000/T [1/K]

[47]

Figure 5.2: Second order rate coefficient for CO2 + Ar → CO + O + Ar: filled squares with error bars, current experimental results; heavy solid line, fit to current data; dashed line, Burmeister and Roth [41]; solid line, Fujii et al. [42]; dash-dot line, Ebrahim and Sandeman [43]; short dash line, Hardy et al. [44]; dash-dot-dot line, Kiefer [46]; smaller dotted line, Dean [47].

3200 3400 3600 3800 4000 4200 4400 4600103

104

105

Z L-J [M

] ∆t in

c

Temperature [K]

Figure 5.3: Number of Lennard-Jones collisions in CO2-Ar incubation period (ZL-J[M]∆tinc): experiment, open circles with error bars; master equation, filled squares.

73

0 50 100 150 2000.5

0.6

0.7

0.8

0.9

1.0

1.1

[CO

2] / [C

O2] 0

Time [µs]

∆ t inc

Figure 5.4: Example master equation result for experiment denoted with an asterisk (*) in Table A.5. Initial conditions: Ttrans = Trot = 4171 K, Tvib =1943 K, and P = 0.735 atm. Open circles, master equation calculation; solid line, fit to calculation for determination of incubation time and rate coefficient.

74

Chapter 6: Benzyl and toluene

decomposition

This chapter discusses the measurement of thermal decomposition rate

coefficients for benzyl decomposition, reaction (7), and toluene decomposition, reactions

(8) and (9). These measurements were made in the shock tube facility using laser

absorption at 266 nm as described in Chapter 2.

6.1 Benzyl decomposition 6.1.1 Determination of benzyl decomposition rate coefficient

Fröchtenicht et al. [117] have shown that the benzyl radical decomposes to yield

an H-atom and a C7H6 fragment of unknown structure

C6H5CH2 → C7H6 + H (7)

The Fröchtenicht et al. [117] experiment utilized photon-induced decomposition of

benzyl in a molecular beam with mass spectrometry detection to identify the C7H6

product. H-atom ARAS measurements [74,87] also indicate that one H-atom is produced

per benzyl undergoing decomposition. Other benzyl decomposition pathways have been

suggested including pathways leading to C5H5 + C2H2 products and C4H4 + C3H3

products [202]. However, the most recent direct high-temperature experimental studies

have concluded that the C7H6 + H pathway is the dominant pathway [74,87,117] and

quantum chemical calculations have confirmed this pathway [80]. In the current study the

decomposition of benzyl, reaction (7), was studied behind reflected shock waves at

temperatures of 1461 to 1730 K and pressures of 1.467 to 1.666 atm using ultraviolet

laser absorption for detection of benzyl at 266 nm. Benzyl was generated by the fast

decomposition of benzyl iodide

C6H5CH2I → C6H5CH2 + I (12)

75

Experiments were performed with mixtures of 50 and 100 ppm benzyl iodide dilute in

argon. The low concentrations provided isolated sensitivity to the decomposition of

benzyl, reaction (7). The progress of reaction was monitored using benzyl absorption at

266 nm. However, interfering absorption due to benzyl fragments and toluene must be

accounted for.

To determine the rate coefficient for reaction (7) the measured 266 nm absorbance

was fit using a detailed reaction mechanism given in Appendix E and the measured

absorption cross-sections for benzyl radicals and the interfering absorbers (benzyl

fragments and toluene). The reaction mechanism was largely developed by Hippler, Troe,

and co-workers [74,81-82,91]. Fortunately, the current low concentration benzyl

decomposition experiments show little sensitivity to secondary chemistry. Therefore,

excellent agreement between fit and experiment was obtained by adjusting only the rate

coefficient for reaction (7). The absorption cross-sections for benzyl and benzyl

fragments were determined in the same experiments used here for the determination of

the benzyl decomposition rate (see section 2.4). The absorption cross-section for toluene

was determined in the toluene decomposition experiments, discussed later in this chapter.

Computations were done using CHEMKIN 2.0 and SENKIN [135-136].

The measured absorbance during an example benzyl decomposition experiment is

shown in Figure 6.1. As described in section 2.4 after the passage of the reflected shock

wave the fast dissociation of benzyl iodide provides an immediate yield of benzyl

radicals and thus the absorption cross-section can be determined. The decay in the

absorbance signal then can be fit by adjusting k7 in the reaction mechanism and

appropriately accounting for absorption due to benzyl fragments, formed via reaction (7),

and toluene formed by benzyl/H-atom recombination

C6H5CH2 +H → C6H5CH3 (14)

The example in Figure 6.1 illustrates the best fit achieved by adjusting k7 and the

contributions to the 266 nm absorbance from the three absorbing species: benzyl, benzyl

fragments, and toluene. As noted in section 2.4, the absorption cross-section of benzyl

fragments was determined from the constant absorption observed at long times and

assuming complete one-to-one conversion of benzyl into benzyl fragments. The

sensitivity for benzyl concentration for the experiment shown in Figure 6.1 is given in

76

Figure 6.2. Notice that the benzyl concentration shows isolated sensitivity to reaction (7)

at early times allowing a direct fit without interference from secondary reactions. The

isolated sensitivity to reaction (7), due to the low concentration of benzyl iodide

precursor, provides a pseudo-first-order fit only complicated by the interfering

absorption. Therefore, uncertainties in the initial concentration of benzyl iodide and the

benzyl absorption cross-section do not affect the uncertainty in the rate coefficient

dermination.

The present rate coefficient determinations for reaction (7) are tabulated in

Appendix A and shown in Figure 6.3 on Arrhenius plot along with a least-squares fit. The

rate coefficient at 1.5 atm given by the least-squares fit can be expressed as

k7(T) = 8.20 x 1014 exp(-40600 K/T) [s-1]

where the RMS experimental scatter about the fit is ±11%. The overall uncertainty of the

rate coefficient is estimated at ±25%. The primary contributions to the uncertainty in the

rate coefficient are uncertainties in: temperature, fitting the data to computed profiles, and

interfering absorption. See Appendix B for details pertaining to the uncertainty analysis.

6.1.2 Comparison with previous studies Comparison of the results of the current study with the results of four previous

shock tube studies is also shown in Figure 6.3. The current data agree very well with the

UV absorption study of Hippler et al. [81] (~0.5 atm) and the H-atom ARAS study of

Braun-Unkhoff et al. [87] (~2 atm); although, the scatter of the current data is

significantly reduced from that of these previous studies. In addition, the current data for

k7 is in agreement with the IUPAC review recommendation, Baulch et al. [201]. (The

IUPAC review recommends the rate coefficient previously determined by Braun-Unkhoff

et al. [87].) The rate coefficient results of the UV absorption study of Jones et al. [80]

(~10-12 atm) and the H-atom ARAS study of Rao and Skinner [86] (~0.6 atm) are

approximately an order-of-magnitude lower than the results of the current study. The

experimental rate coefficient results of Jones et al. [80] and Rao and Skinner [86] have

scatter of approximately an order-of-magnitude and therefore probably have uncertainties

of at least an order-of-magnitude if not greater. The current rate coefficient

determinations have significantly less uncertainty due to the low concentrations used and

77

high sensitivity of the laser absorption technique. Based on the findings for toluene

decomposition (section 6.2), it is expected that k7 is somewhat in the falloff at the highest

temperatures of this study. However, because the C7H6 decomposition product is of

unknown structure (i.e., the transition state is of unknown structure) a RRKM/master

equation calculation cannot be performed to extrapolate the rate coefficient experimental

determinations to the high-pressure-limit.

6.2 Toluene decomposition 6.2.1 Determination of toluene decomposition rate coefficients Mixtures of 200 and 400 ppm of toluene dilute in argon were shock-heated behind

reflected shock waves while absorption at 266 nm, primarily due to benzyl radicals, was

monitored. Reflected shock conditions ranged from 1398 to 1782 K and 1.44 to 1.58 atm.

Toluene is known to decompose via two parallel channels [74,91]

C6H5CH3 → C6H5CH2 + H (8)

C6H5CH3 → C6H5 + CH3 (9)

Therefore, the benzyl absorbance profiles must be fit in terms of the overall

decomposition rate coefficient, k8+k9, and the branching ratio, k8/(k8+k9). The time

dependence of the benzyl sensitivity to the overall decomposition rate coefficient and the

branching ratio allows for the separation of these two parameters. The overall rate of

decomposition, k8+k9, can be determined by fitting the early-time behavior of the benzyl

absorbance profile. At longer times the benzyl absorbance rolls off and the benzyl

sensitivity shifts towards the branching ratio allowing a fit of this parameter.

The measured 266 nm absorbance, see Figure 6.4 for an example, was fit by

adjusting the overall decomposition rate and the branching ratio in the detailed

mechanism given in Appendix E and developed primarily by Hippler, Troe, and co-

workers [74,81-82,91]. As in the case of benzyl decomposition the absorption cross-

sections of benzyl, benzyl fragments, and toluene must be accounted for. The

contribution each of these three absorption components has toward the total signal is

shown in Figure 6.5. The toluene absorption cross-section at 266 nm as determined from

the time-zero absorption during these experiments was found to be σC6H5CH3 = 5.9 (±0.5)

78

x 10-19 cm2molecule-1 with no temperature-dependence for the range of 1398 to 1782 K.

The benzyl and benzyl fragment absorption cross-sections were used as determined in

Chapter 2. A benzyl sensitivity calculation for the conditions of Figure 6.4 is shown in

Figure 6.6. The calculation shows strong sensitivity to the overall decomposition rate,

k8+k9, and the branching ratio, k8/(k8+k9). However the sensitivity calculation also shows

unavoidable interference from fast secondary reactions:

C6H5CH2 → C7H6 + H (7)

C7H6 + H → C6H5CH2 (15)

C6H5CH3 +H → C6H5CH2 + H2 (16)

The rate coefficients for reactions (7) and (16) are well known with uncertainties of

±25% and ±30%, respectively. Reaction (7) is part of the current study and reaction (16)

has been previously measured by Hippler and co-workers [81]. Per the recommendation

of Eng et al. [74], the benzyl fragment C7H6 is assumed to be able to recombine with H to

yield benzyl, reaction (15). The value of k15 was taken from Eng et al. [74] and has large

uncertainty contributing to uncertainty in the determinations for k8 and k9.

The rate coefficient results for toluene decomposition at 1.5 atm are given in

Figures 6.7 and 6.8 with least-squares fits. Expressions for the rate coefficients for

reactions (8) and (9) at 1.5 atm given by the least-squares fits are

k8(T) = 2.09 x 1015 exp(-44040 K/T) [s-1]

k9(T) = 2.66 x 1016 exp(-49260 K/T) [s-1]

where the RMS experimental scatter about the fits is ±13% and ±17% for k8 and k9

respectively. The rate coefficient determinations for reactions (8) and (9) are tabulated in

Appendix A. The primary contributions to uncertainties in the rate coefficients are

uncertainties in: temperature, absorption cross-sections, fitting the data to computed

profiles, and interfering chemistry. These uncertainties give overall uncertainties in k8

and k9 of ±39%, and ±39% respectively. See Appendix B for details of the uncertainty

analysis.

6.2.2 RRKM / master equation calculations Multi-channel RRKM calculations with numerical solution to the 1-D (internal

energy) master equation were carried for the decomposition of toluene. The two

79

transition states were modeled using a restricted Gorin model. The calculations provided

extrapolation of the experimental determinations at 1.5 atm to the high-pressure-limit.

The calculation allowed for a fit of the data with one adjustable parameter: the hindrance,

η, on the restricted internal rotations of the two transition states. The probability function

for collisional energy transfer for deactivating collisions has been accurately measured by

Luther and co-workers [131-132] and their result was used in these calculations. Luther

and co-workers [131-132] give the probability density function using a modified

exponential-down formulation

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

−−=

γ

αα j

ij

jij E

EEN

P10

exp1 for Ej > Ei

where for toluene-argon collisions α0 = 43.5 cm-1, α1 = 4.2 x 10-3, and γ = 0.70. Lennard-

Jones parameters for toluene and argon [140], the molecular frequencies [91,141,165-

167], moments of inertia [91,93,145,166], heats of formation [146,168-171], and barrier

heights [146,168-171] were taken from various references and are listed in Table 6.1.

The RRKM/master equation calculations provided calculation of rate coefficients

in the high-pressure-limit for reactions (8) and (9), for 1398 to 1782 K, which can be

expressed with two-parameter Arrhenius expressions

k∞,8(T) = 8.32 x 1015 exp(-45890 K/T) [s-1]

k∞,9(T) = 1.48 x 1017 exp(-51350 K/T) [s-1]

These expressions along with the RRKM/master equation results for 1.5 atm are shown in

Figures 6.7 and 6.8. Comparison between the least squares fit to the experimental data

and the RRKM/master equation result for 1.5 atm shows excellent agreement.

6.2.3 Comparison with previous studies The results of the current toluene decomposition study, in terms of the overall

decomposition rate coefficient and the branching ratio, are compared to previous studies

in Figures 6.9 and 6.10. The current findings for the overall decomposition rate

coefficient agree, with deviation of less than a factor of two, with previous results of Eng

et al. [74], Braun-Unkhoff et al. [88], and Luther et al. [94]. However the results for the

overall decomposition rate coefficient of Pammidimukkala et al. [89] and Rao and

80

Skinner [90] are significantly lower than the current findings (Figure 6.9). The results of

the current study for the branching ratio are in good agreement with the previous findings

of Eng et al. [74] and Luther et al. [94]. However the results of the current study are in

disagreement with the results of Pammidimukkala et al. [89], Rao and Skinner [90], and

Braun-Unkhoff et al. [87]. The current data set and the H-atom ARAS experiments of

Eng et al. [74] clearly show that the toluene decomposition channel leading to benzyl is

more dominant than the channel leading to phenyl. On these grounds the low branching

ratios given by the earlier Pammidimukkala et al. [89] (1987) and Rao and Skinner [90]

(1989) are probably erroneous. Additionally the relatively low uncertainties of the current

results and the good agreement with the low concentration (high kinetic isolation) H-

atom ARAS results of Eng et al. [74] lends confidence to the results presented here.

A comparison of the toluene decomposition results with the most recent IUPAC

recommendations (Baulch et al. [201]) is also made in Figures 6.7-6.10. The IUPAC

recommendation for the toluene → benzyl + H channel, k8, shows good agreement.

However, the recommendation for the toluene → phenyl + CH3 channel, k9, shows large

disagreement. The IUPAC recommendation has far too low an activation energy and A-

factor in comparison to the current study and precious studies. This disagreement results

in disagreement in the magnitude of the branching ratio and the temperature-dependence,

Figure 6.10.

81

Table 6.1: Parameters used in RRKM / master equation calculations for toluene

decomposition.

C6H5CH3Heat of Formation (0K) [168]: 17.50 kcal/mol Moments of Inertia (amu Å2) [93]: 290.7 (inactive); 200.7 (inactive); 89.2 (active), σ = 2 Frequencies (cm-1) [165]: 3085, 3070, 3058, 3037, 3028, 2979, 2950, 2920,

1604, 1584, 1493, 1455(3), 1378, 1331, 1313, 1208, 1176, 1153, 1080, 1040(2), 1028, 1002, 983, 973, 893, 841, 784, 734, 690, 620, 524, 467, 406, 347, 217

Internal rotation (amu Å2) [167]: 3.139 C6H5CH2 … H

Frequencies (cm-1) [91]: 3060(2), 3050, 3040, 3030, 2930, 2860, 1600, 1550, 1480, 1440(2), 1350, 1330, 1300, 1270, 1160(2), 1070, 1010, 980(2), 950, 890, 820, 810, 750, 640, 610, 600, 520, 420, 360(2), 290, 170

Active 1-D external rotor (amu Å2) [93]: 89.2, σ = 2 Active 2-D internal rotor (amu Å2) [91]: 186.07, σ = 1; η = 99.84% Active 2-D internal rotor (amu Å2) [91]: 275.00, σ = 1; η = 99.84%Barrier at 0 K [168-169,171,204]: 88.86 kcal/mol

C6H5 … CH3Frequencies (cm-1) [141,166]: 3085, 3071, 3052, 3073, 3060, 1593, 1499, 1441,

1433, 1344, 1226, 1086, 1080, 1067, 1027, 1011, 976, 971, 966, 878, 813, 708, 629, 605, 586, 416, 400, 3184(2), 3002, 1383(2), 580

Active 1-D external rotor (amu Å2) [93]: 89.2, σ = 2 Active 1-D internal torsion (amu Å2) [145,166]: 3.421, σ = 3Active 2-D internal rotor (amu Å2) [145]: 1.760, σ = 2; η = 82% Active 2-D internal rotor (amu Å2) [145]: 1.760, σ = 2; η = 82% Active 2-D internal rotor (amu Å2) [166]: 80.66, σ = 1; η = 82% Active 2-D internal rotor (amu Å2) [166]: 172.02, σ = 1; η = 82%Barrier at 0 K [146,168,170]: 102.58 kcal/mol

82

-100 0 100 200 300 400 500

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Abs

orba

nce

Time [µs]

Figure 6.1: Example 266 nm absorbance during benzyl iodide decomposition. Reflected shock conditions: 1615 K, 1.556 atm, 50 ppm benzyl iodide/Ar. Solid line, best fit to absorbance by adjusting k7; dashed line, contribution to absorbance from benzyl; dotted line, contribution to absorbance from benzyl fragments; dot-dashed line, contribution to absorbance from toluene.

0 50 100 150 200-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Loca

l Ben

zyl S

ensi

tivity

, S

Time [µs]

C6H5CH2=>C7H6+H C6H5CH3=>C6H5CH2+H C6H5CH2+H=>C6H5CH3 C6H5CH3+H=>C6H5CH2+H2

Figure 6.2: Local sensitivity for benzyl concentration for conditions of experiment given in Figure 6.1. S = (dXbenzyl / dki) (ki / Xbenzyl,local), where ki is the rate constant for reaction i and Xbenzyl,local is the local benzyl (C6H5CH2) mole fraction.

83

0.55 0.60 0.65 0.70101

102

103

104

105

k 7 [1/s

]

1000/T [1/K]

Figure 6.3: Rate coefficient for benzyl decomposition, reaction (7): filled squares with error bars, current experimental results (~1.5 atm); heavy solid line, fit to current data; dash-dot-dot line, Hippler et al. [81] (~0.5 atm); dashed line, Braun-Unkhoff et al. [87] (~2 atm) and Baulch et al. [201]; dotted line, Jones et al. [80] (~10-12 atm); dash-dot line, Rao and Skinner [86] (~0.6 atm).

-100 0 100 200 300 400 500

0.00

0.02

0.04

0.06

0.08

Abs

orba

nce

Time [µs]

Figure 6.4: Example 266 nm absorbance during toluene decomposition. Reflected shock conditions: 1593 K, 1.461 atm, 200 ppm toluene/Ar. Solid line, fit to data by adjusting overall decomposition rate, k8+k9, and branching ratio, k8/(k8+k9); dotted lines, variation of k8+k9 ±50%; dashed lines, variation of k8/(k8+k9) ±30%.

84

-100 0 100 200 300 400 500

0.00

0.02

0.04

0.06

0.08

Abs

orba

nce

Time [µs]

Figure 6.5: Contribution to 266 nm absorbance from benzyl (dashed line), benzyl fragment (dashed-dotted line), and toluene (dotted line) for example experiment given in Figure 6.4.

0 100 200 300 400 500

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Loca

l Ben

zyl S

ensi

tivity

, S

Time [µs]

k8 + k9

k8 / (k8 + k9) C6H5CH3+H=>C6H5CH2+H2 C7H6+H=>C6H5CH2 C6H5CH2=>C7H6+H

Figure 6.6: Local sensitivity for benzyl concentration for conditions of experiment given in Figure 6.4. S = (dXbenzyl / dki) (ki / Xbenzyl,local), where ki is the rate constant for reaction i and Xbenzyl,local is the local benzyl (C6H5CH2) mole fraction.

85

0.56 0.63 0.70101

102

103

104

105

k 8 [1/s

]

1000/T [1/K]

Figure 6.7: Rate coefficient for C6H5CH3 → C6H5CH2 + H, reaction (8): filled squares with error bars, current experimental results (~1.5 atm); solid gray line, fit to current data; solid black line, RRKM / master equation result for k8,∞; dashed line, RRKM / master equation result for k8(1.5 atm); dotted line, Baulch et al. [201] IUPAC recommendation for k8,∞.

0.55 0.60 0.65 0.70 0.75100

101

102

103

104

105

k 9 [1/s

]

1000/T [1/K]

Figure 6.8: Rate coefficient for C6H5CH3 → C6H5 + CH3, reaction (9): filled circles with error bars, current experimental results (~1.5 atm); solid gray line, fit to current data; solid black line, RRKM / master equation result for k9,∞; dashed line, RRKM / master equation result for k9(1.5 atm); dotted line, Baulch et al. [201] IUPAC recommendation for k9,∞.

86

0.50 0.55 0.60 0.65 0.70 0.75 0.80101

102

103

104

105

106

k 8 + k

9 [1/s

]

1000/T [1/K]

Oehlschlaeger (this study), 1.5 atm Pamidimukkala et al. [89], 0.5 atm Rao and Skinner [90], 0.4 atm Luther et al. [94], infinite p Eng et al. [74], 2 atm Braun-Unkhoff et al. [88], 2 atm Baulch et al. [201], infinite p

Figure 6.9: Comparison with previous data for the toluene overall decomposition rate coefficient k8 + k9. Luther et al. [94] performed laser excitation study to determine rate coefficient at infinite pressure.

1200 1400 1600 1800 20000.0

0.2

0.4

0.6

0.8

1.0

k 8 / (k

8 + k

9)

Temperature [K]

Oehlschlaeger (this study), 1.5 atm Pamidimukkala et al. [89], 0.5 atm Rao and Skinner [90], 0.4 atm Braun-Unkhoff et al. [88], 2 atm Luther et al. [94], infinite p Eng et al. [74], 2 atm Baulch et al. [201], infinite p

Figure 6.10: Comparison with previous data for the toluene decomposition branching ratio k8 / (k8 + k9). Luther et al. [94] performed laser excitation study to branching ratio at infinite pressure.

87

Chapter 7: Conclusions

7.1 Summary of results The objective of the research presented in this thesis was to make accurate

measurements of rate coefficients for several decomposition reactions of importance in

combustion systems. The topics are broken into three categories: 1) the decomposition of

alkanes, 2) the incubation and decomposition of CO2, and 3) the decomposition of benzyl

and toluene. Sensitive ultraviolet laser absorption for detection of product or reactant

species was used to monitor the progress of reaction in shock-wave initiated thermal

decomposition experiments. The rate coefficient and incubation results were fit using

computational techniques based on RRKM theory and the one-dimensional master

equation.

7.1.1 Alkane decomposition Laser absorption of methyl radicals at 216.62 nm provided a sensitive, high-

accuracy method to determine the decomposition rates for ethane, propane, iso-butane,

and n-butane:

C2H6 → CH3 + CH3 (1)

C3H8 → CH3 + C2H5 (2)

i-C4H10 → CH3 + i-C3H7 (3)

n-C4H10 → CH3 + n-C3H7 (4)

n-C4H10 → C2H5 + C2H5 (5)

Using this technique measurements of k1-k5 were made in the falloff regime with

experimental conditions ranging from 1297 to 2034 K and 0.13 to 8.8 atm [172-173].

Expressions for the high- and low-pressure-limit rate coefficients and Troe-formulated

falloff parameters for reactions (1)-(5) were determined from RRKM/master equation

calculations fit to the experimental data. The reaction (1) rate expressions for 700 to 1924

K are:

89

k∞,1(T) = 1.88 x 1050 T -9.72 exp(-54020 K/T) [s-1]




k∞,2(T) = 1.29 x 1037 T -5.84 exp(-49010 K/T) [s-1]




k∞,3(T) = 4.83 x 1016 exp(-40210 K/T) [s-1]

k0,3(T) = 2.41 x 1019 exp(-26460 K/T) [cm3mol-1s-1]

Fcent,3(T) = 0.75 exp(-T/750 K)


k∞,4(T) = 4.28 x 1014 exp(-35180 K/T) [s-1]

k0,4(T) = 5.34 x 1017 exp(-21620 K/T) [cm3mol-1s-1]

Fcent,4(T) = 0.28 exp(-T/1500 K)

And the reaction (5) rate expressions for 1320 to 1600 K are:

k∞,5(T) = 2.72 x 1015 exp(-38050 K/T) [s-1]

k0,5(T) = 4.72 x 1018 exp(-24950 K/T) [cm3mol-1s-1]

Fcent,5(T) = 0.28 exp(-T/1500 K)

The estimated uncertainties in k1-k5 are ±19%, ±26%, ±25%, ±31%, and ±31%

respectively. This is the first study in which several of these reactions have been studied

using adequately dilute mixtures in order to sufficiently isolate the decomposition

reactions and the results for k5 represent the first high-temperature measurement of this

rate coefficient. Additionally, these rate determinations provide a test of validity for the

geometric mean rule used for rate coefficient estimation. The current data set does in fact

suggest that the geometric mean rule is a good estimator of decomposition or association

rate coefficients.

7.1.2 CO2 incubation and decomposition Incubation prior to the thermal decomposition of CO2

CO2 + M → CO + O(3P) + M (6)

90

was observed for the first time in reflected shock wave experiments, confirming the

expected bottleneck in collisional activation of CO2 [174]. The thermal decomposition of

carbon dioxide was investigated behind reflected shock waves at temperatures of 3200 to

4600 K and pressures of 0.44 to 0.98 atm. Ultraviolet laser absorption at 216.5 and 244

nm was used to monitor the CO2 concentration with microsecond time resolution. The

second order rate coefficient for CO2 dissociation was measured with good agreement

with several previous shock tube studies. The second-order rate coefficient for CO2

decomposition, reaction (6), over the temperature range of 3200 to 4600 K and around

0.5 to 1 atm is given by

k6(T) = 3.14 x 1014 exp(-51300 K/T) [cm3mol-1s-1]

with estimated uncertainty of ±21%. The number of incubation collisions was found to

range from 7 x 103 at 4600 K to 3.5 x 104 at 3200 K. Master equation simulations, with a

simple model for collisional energy transfer, were carried out to describe the measured

incubation times and unimolecular rate coefficient. It was found that the experimental

incubation times could not be simulated without a collisional energy transfer parameter,

α, that increased with a greater than linear dependence on energy. An exponential model

for the energy transfer probability density function, P(E,E’), with a quadratic energy

dependence for α was found to adequately represent the experimental incubation times.

7.1.3 Benzyl and toluene decomposition The decomposition of benzyl radicals and toluene

C6H5CH2 → C7H6 + H (7)

C6H5CH3 → C6H5CH2 + H (8)

C6H5CH3 → C6H5 + CH3 (9)

was studied at high-temperatures behind shock waves using laser absorption of benzyl

radicals at 266 nm to monitor the progress of reaction. The rate coefficient for benzyl

decomposition, reaction (7), determined at 1.5 atm and 1461 to 1730 K is given by

k7(T) = 8.20 x 1014 exp(-40600 K/T) [s-1]

and the rate coefficients for the two-channel toluene decomposition, determined at 1.5

atm and 1398 to 1782 K are given by

k8(T) = 2.09 x 1015 exp(-44040 K/T) [s-1]

91

k9(T) = 2.66 x 1016 exp(-49260 K/T) [s-1]

Additionally, the toluene decomposition rate coefficient results were extrapolated to the

high-pressure-limit with RRKM/master equation calculations, the resulting rate

expressions for 1398 to 1782 are given by

k∞,8(T) = 8.32 x 1015 exp(-45890 K/T) [s-1]

k∞,9(T) = 1.48 x 1017 exp(-51350 K/T) [s-1]

The results for k7-k9 are in agreement with several previous experimental studies

while in disagreement with others. The low concentrations used and the high levels of

signal-to-noise provided by the laser absorption technique allowed for the determination

of k7-k9 with small uncertainties relative to previous studies, ±25%, ±39%, and ±39% in

k7-k9 respectively. These results put firm boundaries on the values these rate coefficients

of significant importance in describing toluene combustion.

7.2 Recommendations for future work 7.2.1 High-pressure ethane and propane decomposition

measurements The current measurements for ethane and propane decomposition are a bit far

from the high-pressure-limit to adequately judge theoretical calculations of the high-

pressure rate coefficient. The ethane measurements are a factor of ten from the high-

pressure limit at 1500 K and the measurements for propane decomposition are a factor of

four from the high-pressure limit at 1500 K. Measurements around 100 atm, possible in

the Stanford High Pressure Shock Tube facility, would be much closer to the high-

pressure-limit than the current data set and would provide insight into the amount of

high-temperature roll-off in the high-pressure rate coefficient. However, performing these

experiments would not be trivial because of inherent difficulties of laser absorption at

high-pressure, see Ph.D. thesis by E.L Petersen [175].

7.2.2 Oxidation of toluene Due to its importance as a representative aromatic fuel, toluene has begun to

receive attention in the combustion modeling community; consequently there are several

92

comprehensive oxidation mechanisms under development [75-79]. In addition, many of

the reaction rate coefficients of importance are poorly known. One of these reactions is

the benzyl + O2 reaction. At high-temperatures the decomposition of toluene initiates

toluene oxidation. Therefore, the benzyl + O2 reaction is an important reaction in the

early stages of toluene oxidation. Measurement of the benzyl + O2 rate coefficient could

be made by monitoring benzyl, at 266 nm, in shock-heated mixtures of small

concentrations of benzyl iodide in an excess of oxygen dilute in argon. The benzyl iodide

would immediately decompose behind the reflected shock wave leaving benzyl which

would react in a pseudo first-order manner with the excess oxygen. Of course this

reaction would compete with the benzyl decomposition reaction measured as part of the

work of this thesis.

7.2.2 HO2 and H2O2 reactions It is known that during hydrogen and hydrocarbon oxidation at lower

temperatures (~700-1200 K) the chain branching mechanism changes from the familiar

high-temperature mechanism, rate limited by H + O2 → OH + O, to one in which

reactions involving hydrogen peroxide (H2O2) and the hydroperoxy radical (HO2) control

chain branching. However, at these temperatures the rate coefficients of many important

reactions involving HO2 and H2O2 have large uncertainties. New measurements of these

rate coefficients with smaller uncertainties are needed for the accurate modeling of these

systems. Additionally, an understanding of low-temperature hydroperoxy chemistry is

needed before one can attack the larger problem of peroxy (RO2) chemistry important in

low-temperature large hydrocarbon oxidation.

Fortunately, the HO2 radical has strong absorption from 200 to 230 nm and can be

detected using the 216 nm laser setup developed as part of this thesis for methyl radical

measurements. Additionally, the hydroxyl radical (OH) is an important species in the

reactions of interest and can be detected using laser absorption at 306 nm. The following

reactions show strong sensitivity in low- to moderate-temperature combustion and are

candidates to be studied using HO2 and OH laser absorption:

H + H2O2 → HO2 + H2 (17)

HO2 + HO2 → O2 + H2O2 (18)

93

OH + H2O2 → HO2 + H2O (19)

OH + HO2 → H2O + O2 (20)

Accurate measurements of rate coefficients for reactions (17)-(20) would be a significant

contribution to the combustion community.

94

Appendix A: Rate coefficient data

Table A.1: Summary of experimental results for C2H6 decomposition. Mixture Temperature [K] Pressure [atm] k1 [1/s]

398 ppm C2H6 / Ar 1503 0.219 4.2 x 102 * 398 ppm C2H6 / Ar 1566 0.207 8.2 x 102 * 398 ppm C2H6 / Ar 1623 0.196 1.6 x 103 * 398 ppm C2H6 / Ar 1720 0.181 4.3 x 103 * 398 ppm C2H6 / Ar 1795 0.167 7.5 x 103 * 398 ppm C2H6 / Ar 1924 0.148 1.8 x 104 * 398 ppm C2H6 / Ar 2034 0.126 2.7 x 104 * 202 ppm C2H6 / Ar 1862 1.511 5.2 x 104

202 ppm C2H6 / Ar 1716 1.576 1.4 x 104

202 ppm C2H6 / Ar 1589 1.664 3.7 x 103

202 ppm C2H6 / Ar 1488 1.730 7.7 x 102

100 ppm C2H6 / Ar 1891 1.432 7.0 x 104

402 ppm C2H6 / Ar 1421 1.859 2.8 x 102

402 ppm C2H6 / Ar 1435 1.600 3.9 x 102

200 ppm C2H6 / Ar 1624 4.190 8.3 x 103

200 ppm C2H6 / Ar 1537 4.326 2.8 x 103

200 ppm C2H6 / Ar 1453 4.434 7.5 x 102

200 ppm C2H6 / Ar 1676 4.204 1.5 x 104

200 ppm C2H6 / Ar 1753 3.847 4.2 x 104

101 ppm C2H6 / Ar 1419 4.493 4.2 x 102

101 ppm C2H6 / Ar 1489 4.391 1.7 x 103

402 ppm C2H6 / Ar 1859 3.885 9.5 x 104

100 ppm C2H6 / Ar 1710 7.068 3.5 x 104

100 ppm C2H6 / Ar 1598 7.620 9.0 x 103

100 ppm C2H6 / Ar 1494 7.983 1.7 x 103

100 ppm C2H6 / Ar 1771 6.942 5.7 x 104

402 ppm C2H6 / Ar 1504 8.059 2.0 x 103

* indicates incident shock experiments

95

Table A.2: Summary of experimental results for C3H8 decomposition. Mixture Temperature [K] Pressure [atm] k2 [1/s]

399 ppm C3H8 / Ar 1435 0.228 1.3 x 103 * 399 ppm C3H8 / Ar 1365 0.244 4.5 x 102 * 399 ppm C3H8 / Ar 1572 0.206 7.5 x 103 * 399 ppm C3H8 / Ar 1653 0.187 1.7 x 104 * 201 ppm C3H8 / Ar 1662 1.561 6.5 x 104

201 ppm C3H8 / Ar 1553 1.616 1.9 x 104

201 ppm C3H8 / Ar 1448 1.683 3.9 x 103

201 ppm C3H8 / Ar 1379 1.759 1.2 x 103

402 ppm C3H8 / Ar 1375 1.772 1.2 x 103

201 ppm C3H8 / Ar 1554 4.378 2.5 x 104

201 ppm C3H8 / Ar 1433 4.535 3.6 x 103

201 ppm C3H8 / Ar 1538 3.715 2.1 x 104

201 ppm C3H8 / Ar 1658 3.529 8.9 x 104

201 ppm C3H8 / Ar 1550 4.387 2.3 x 104

402 ppm C3H8 / Ar 1426 4.514 3. 1x 103

402 ppm C3H8 / Ar 1343 4.712 9.0 x 102

101 ppm C3H8 / Ar 1490 7.928 1.0 x 104

101 ppm C3H8 / Ar 1414 8.092 3.3 x 103

101 ppm C3H8 / Ar 1562 7.757 3.2 x 104

101 ppm C3H8 / Ar 1607 7.548 5.6 x 104

200 ppm C3H8 / Ar 1387 8.393 2.2 x 103

* indicates incident shock experiments Table A.3: Summary of experimental results for i-C4H10 decomposition.

Mixture Temperature [K] Pressure [atm] k3 [1/s] 400 ppm i-C4H10 / Ar 1352 0.251 1.8 x 103 * 400 ppm i-C4H10 / Ar 1401 0.238 4.4 x 103 * 400 ppm i-C4H10 / Ar 1458 0.224 9.5 x 103 * 400 ppm i-C4H10 / Ar 1539 0.212 2.8 x 104 * 201 ppm i-C4H10 / Ar 1417 1.731 1.1 x 104

201 ppm i-C4H10 / Ar 1365 1.785 4.5 x 103

201 ppm i-C4H10 / Ar 1310 1.815 1.3 x 103

201 ppm i-C4H10 / Ar 1468 1.623 2.8 x 104

201 ppm i-C4H10 / Ar 1483 1.660 3.2 x 104

201 ppm i-C4H10 / Ar 1551 1.607 8.0 x 104

200 ppm i-C4H10 / Ar 1395 4.318 9.0 x 103

200 ppm i-C4H10 / Ar 1392 4.510 9.0 x 103

200 ppm i-C4H10 / Ar 1344 4.480 3.0 x 103

200 ppm i-C4H10 / Ar 1469 4.432 3.4 x 104

200 ppm i-C4H10 / Ar 1522 4.244 7.2 x 104

200 ppm i-C4H10 / Ar 1414 8.218 1.5 x 104

200 ppm i-C4H10 / Ar 1368 8.420 6.3 x 103

200 ppm i-C4H10 / Ar 1333 8.843 3.1 x 103

200 ppm i-C4H10 / Ar 1535 7.934 1.2 x 105

* indicates incident shock experiments

96

Table A.4: Summary of experimental results for n-C4H10 decomposition. Mixture Temperature [K] Pressure [atm] k4 [1/s] k5 [1/s]

399 ppm n-C4H10 / Ar 1337 0.250 7.0 x 102 6.0 x 102 * 399 ppm n-C4H10 / Ar 1432 0.226 3.5 x 103 2.0 x 103 * 399 ppm n-C4H10 / Ar 1500 0.213 8.5 x 103 6.8 x 103 * 399 ppm n-C4H10 / Ar 1560 0.203 1.5 x 104 1.2 x 104 * 399 ppm n-C4H10 / Ar 1381 0.235 1.5 x 103 1.2 x 103 * 200 ppm n-C4H10 / Ar 1479 1.673 1.3 x 104 1.3 x 104

200 ppm n-C4H10 / Ar 1601 1.677 4.5 x 104 4.5 x 104

200 ppm n-C4H10 / Ar 1436 1.752 5.5 x 103 4.8 x 103

200 ppm n-C4H10 / Ar 1344 1.869 9.0 x 102 6.0 x 102

200 ppm n-C4H10 / Ar 1541 1.659 2.3 x 104 2.2 x 104

200 ppm n-C4H10 / Ar 1360 1.786 1.4 x 103 1.0 x 103

198 ppm n-C4H10 / Ar 1376 4.282 2.5 x 103 2.0 x 103

198 ppm n-C4H10 / Ar 1447 4.303 8.0 x 103 8.0 x 103

198 ppm n-C4H10 / Ar 1404 3.778 4.3 x 103 3.0 x 103

198 ppm n-C4H10 / Ar 1520 4.208 3.0 x 104 3.0 x 104

200 ppm n-C4H10 / Ar 1414 8.183 5.0 x 103 5.0 x 103

200 ppm n-C4H10 / Ar 1344 8.340 1.3 x 103 1.0 x 103

200 ppm n-C4H10 / Ar 1297 8.411 5.0 x 102 3.0 x 102

200 ppm n-C4H10 / Ar 1479 8.064 1.4 x 104 1.2 x 104

* indicates incident shock experiments Table A.5: Summary of experimental results for CO2 decomposition and incubation.

P2 [atm] T2 [K] P5 [atm] T5 [K] k6 [cm3 mol-1 s-1] ∆tinc [µs] 2% CO2 / Ar

0.198 1547 0.975 3209 3.4 x 107 24 0.186 1624 0.927 3393 9.0 x 107 25 0.189 1627 0.944 3399 1.0 x 108 25 0.171 1722 0.866 3624 2.1 x 108 26 0.170 1761 0.862 3717 3.4 x 108 23 0.157 1802 0.800 3814 4.0 x 108 23 0.161 1812 0.820 3838 4.9 x 108 22 0.146 1889 0.753 4021 9.4 x 108 10 0.131 1999 0.684 4281 2.1 x 109 9 0.084 2088 0.443 4494 3.6 x 109 23

1% CO2 / Ar 0.193 1607 0.950 3369 9.0 x 107 28 0.187 1629 0.924 3420 9.5 x 107 31 0.179 1686 0.890 3558 1.8 x 108 20 0.175 1723 0.876 3645 2.1 x 108 25 0.170 1725 0.847 3650 2.6 x 108 20 0.167 1767 0.837 3751 3.2 x 108 15 0.156 1837 0.788 3918 6.6 x 108 12 0.144 1867 0.732 3989 7.6 x 108 14 0.144 1943 0.735 4171 1.6 x 109 13* 0.139 1955 0.713 4200 1.5 x 109 14 0.136 2022 0.701 4358 2.3 x 109 12 0.131 2037 0.676 4394 2.7 x 109 10 0.128 2074 0.664 4484 3.6 x 109 12 0.123 2123 0.637 4601 4.8 x 109 11

2 - incident shock conditions 5 - reflected shock conditions * - see Figure 5.4 for master equation calculation

97

Table A.6: Summary of experimental results for benzyl radical decomposition. Mixture Temperature [K] Pressure [atm] k7 [1/s]

100 ppm benzyl iodide / Ar 1715 1.484 3.9 x 104
















Table A.7: Summary of experimental results for toluene decomposition. Mixture Temperature [K] Pressure [atm] k8 [1/s] k9 [1/s]

100 ppm toluene / Ar 1643 1.516 6.0 x 103 3.1 x 103

100 ppm toluene / Ar 1587 1.560 2.1 x 103 1.1 x 103

100 ppm toluene / Ar 1555 1.562 1.3 x 103 5.0 x 102

200 ppm toluene / Ar 1580 1.508 1.7 x 103 7.0 x 102

200 ppm toluene / Ar 1593 1.461 1.9 x 103 1.0 x 103

200 ppm toluene / Ar 1782 1.438 3.6 x 104 2.0 x 104

200 ppm toluene / Ar 1497 1.580 3.4 x 102 1.2 x 102

400 ppm toluene / Ar 1562 1.527 1.2 x 103 7.0 x 102

400 ppm toluene / Ar 1429 1.518 8.0 x 101 2.5 x 101

400 ppm toluene / Ar 1647 1.431 4.5 x 103 3.0 x 103

400 ppm toluene / Ar 1398 1.504 4.0 x 101 1.2 x 101

400 ppm toluene / Ar 1745 1.484 2.0 x 104 1.3 x 104

98

Appendix B: Uncertainty analysis

The uncertainty estimates for the rate coefficient determinations for reactions (1)-

(9), given in Chapters 4-6, were determined by considering the combination of several

possible sources of error. Assessment of the uncertainty in rate coefficient determinations

was performed for individual experiments representing the extreme temperature limits for

each data set. The limits of the temperature range for each data set usually represent the

maximum uncertainty conditions for that data set with the lowest temperature

experiments typically exhibiting the greatest uncertainty for decomposition experiments.

The error sources considered for each rate coefficient are given in Tables B.2-B.10 for

reactions (1)-(9) for the experimental condition which represents the condition of

maximum uncertainty for each data set. The primary contributions to uncertainties in the

rate coefficients considered are uncertainties in: temperature, absorption cross-sections,

fitting the data to computed profiles, and uncertainties resulting from secondary

chemistry. The individual contributions of each error source were determined by

perturbing each error source to the estimated positive and negative bounds of its 2-σ

uncertainty and re-fitting the experimental traces by adjusting the rate coefficient of

interest. The resulting uncertainty in the rate coefficient for each individual error source

were combined using the root-sum-squares (RSS) method, which assumes that the

uncertainty contributions are uncorrelated and are estimated to similar probabilities (e.g.,

2-σ probability that the true value falls within the ± error limit). Table B.1 gives the

estimated uncertainties for the rate coefficients for reactions (1)-(9), respectively. In the

case of two-channel reactions, n-butane and toluene, the uncertainty in the overall rate of

decomposition and branching ratio are combined using the RSS method to determine the

uncertainty in the two different branches.

99

Table B.1: Uncertainties in k1-k9. Reaction 2-σ estimated uncertainty (1) C2H6 → CH3 + CH3 ±19% (2) C3H8 → CH3 + C2H5 ±26% (3) iC4H10 → CH3 + iC3H7 ±25% (4) nC4H10 → CH3 + nC3H8 ±31% (5) nC4H10 → C2H5 + C2H5 ±31% (6) CO2 + Ar → CO + O + Ar ±21% (7) C6H5CH2 → C7H6 + H ±25% (8) C6H5CH3 → C6H5CH2 + H ±39% (9) C6H5CH3 → C6H5 + CH3 ±39% Table B.2: Uncertainty analysis for ethane decomposition rate coefficient, k1; initial reflected shock conditions: 1421 K, 1.859 atm, and 402 ppm C2H6/Ar.

Error source 2-σ uncertainty in error source

Uncertainty in k1 from positive perturbation

Uncertainty in k1 from negative perturbation

2-σ average uncertainty contribution

Experimental uncertaintiesTemperature and pressure ± 0.7%, ± 1% -16% 19% 17.5% Fitting (signal/noise) ± 5% 5% -5% 5% CH3 absorption coefficient ± 5% -5% 5% 5% Mixture concentration ± 0.43 ppm C2H6 >0.5% >0.5% >0.5% Secondary chemistry uncertaintiesNo sensitive interfering reactions Total 2-σ R.S.S. uncertainty: ±19% Table B.3: Uncertainty analysis for propane decomposition rate coefficient, k2; initial incident shock conditions: 1365 K, 0.244 atm, and 399 ppm C3H8/Ar.





Experimental uncertaintiesTemperature and pressure ± 0.7%, ± 1% -17% 20% 18.5% Fitting (signal/noise) ± 5% 5% -5% 5% CH3 absorption coefficient ± 5% -5% 5% 5% Mixture concentration ± 0.43 ppm C3H8 >0.5% >0.5% >0.5% Secondary chemistry uncertaintiesH+C3H8 → C3H7 + H2 x2, ÷2 -14% 20% 17% Total 2-σ R.S.S. uncertainty: ±26%

100

Table B.4: Uncertainty analysis for iso-butane decomposition rate coefficient, k3; initial reflected shock conditions: 1310 K, 1.815 atm, and 201 ppm i-C4H10/Ar.





Experimental uncertaintiesTemperature and pressure ± 0.7%, ± 1% -19% 22% 20.5% Fitting (signal/noise) ± 5% 5% -5% 5% CH3 absorption coefficient ± 5% -5% 5% 5% Mixture concentration ± 0.86 ppm

iC4H10

>0.5% >0.5% >0.5%

Interfering absorption ± 10% in resulting XCH3

-10% 10% 10%

Secondary chemistry uncertaintiesH+i-C4H10 → i-C4H9 + H2 x2, ÷2 -7% 10% 8.5% Total 2-σ R.S.S. uncertainty: ±25% Table B.5: Uncertainty analysis for overall n-butane decomposition rate coefficient, k4 + k5; initial incident shock conditions: 1337 K, 0.250 atm, and 399 ppm n-C4H10/Ar.


Uncertainty in k4 + k5 from positive

perturbation

Uncertainty in k4 + k5 from negative

perturbation



nC4H10

>0.5% >0.5% >0.5%

Secondary chemistry uncertaintiesn-C4H10 + H → H2 + sC4H9 x2, ÷2 -5% 7% 6% CH3 + CH3 → C2H6 ±19% 0% 0% 0% Total 2-σ R.S.S. uncertainty: ±24% Table B.6: Uncertainty analysis for n-butane decomposition branching ratio, k4/(k4 + k5) ; initial incident shock conditions: 1337 K, 0.250 atm, and 399 ppm n-C4H10/Ar.

Error source 2-σ

uncertainty in error source

Uncertainty in k4/(k4 + k5) from

positive perturbation

Uncertainty in k4/(k4 + k5) from

negative perturbation



nC4H10

>0.5% >0.5% >0.5%

Secondary chemistry uncertaintiesn-C4H10 + H → H2 + sC4H9 x2, ÷2 -3% 4% 3.5% CH3 + CH3 → C2H6 ±19% 17% -15% 16% Total 2-σ R.S.S. uncertainty: ±20%

101

Table B.7: Uncertainty analysis for CO2 decomposition rate coefficient, k6; initial reflected shock conditions: 3209 K, 0.975 atm, and 2% CO2/Ar.





Experimental uncertaintiesInitial temperature and pressure

± 0.7%, ± 1% -12% 13% 12.5%

Temperature and pressure change due to decomposition

± 15% 15% -15% 15%

Fitting (signal/noise) ± 5% 5% -5% 5% CO2 absorption cross-section

± 5% -5% 5% 5%

Mixture concentration ± 0.13% in XCO2 >0.2% >0.2% >0.2% Secondary chemistry uncertaintiesNo sensitive interfering reactions Total 2-σ R.S.S. uncertainty: ±21% Table B.8: Uncertainty analysis for benzyl decomposition rate coefficient, k7; initial reflected shock conditions: 1428 K, 1.649 atm, and 50 ppm benzyl iodide/Ar.

Error source 2-σ

uncertainty in error source




Experimental uncertaintiesTemperature and pressure ± 0.7%, ± 1% -20% 24% 22% Fitting (signal/noise) ± 5% 5% -5% 5% Benzyl absorption cross-section

± 20% 0% (due to pseudo-1st-order conditions)

0% 0%

Interfering species absorption cross-sections

± 20% -10% 10% 10%

Mixture concentration ± 5 ppm benzyl iodide

0% (due to pseudo-1st-order conditions)

0% 0%

Secondary chemistry uncertaintiesC6H5CH3 → C6H5CH2 + H ± 30% 0% 0% 0% Total 2-σ R.S.S. uncertainty: ±25%

102

Table B.9: Uncertainty analysis for overall toluene decomposition rate coefficient, k8 + k9; initial reflected shock conditions: 1429 K, 1.518 atm, and 400 ppm toluene/Ar.



perturbation


perturbation


Experimental uncertaintiesTemperature and pressure ± 0.7%, ± 1% -21% 27% 24% Fitting (signal/noise) ± 5% 5% -5% 5% Benzyl absorption cross-section

± 20% -15% 15% 15%


± 20% -5% 5% 5%

Mixture concentration ± 20 ppm toluene

-5% 5% 5%

Secondary chemistry uncertaintiesC6H5CH3 +H → C6H5CH2 + H2

± 50% -11% 13% 12%

C7H6 + H → C6H5CH2 x2, ÷10 0 6% 3% C6H5CH2 → C7H6 + H ± 25% 3% -2% 2.5% Total 2-σ R.S.S. uncertainty: ±32% Table B.10: Uncertainty analysis for overall toluene decomposition branching ratio, k8 / (k8 + k9) ; initial reflected shock conditions: 1429 K, 1.518 atm, and 400 ppm toluene/Ar.



perturbation


perturbation


Experimental uncertaintiesTemperature and pressure ± 0.7%, ± 1% -2% 3% 2.5% Fitting (signal/noise) ± 5% 5% -5% 5% Benzyl absorption cross-section

± 20% -10% 10% 10%


± 20% -5% 5% 5%

Mixture concentration ± 20 ppm toluene

-5% 5% 5%

Secondary chemistry uncertaintiesC6H5CH3 +H → C6H5CH2 + H2

x2, ÷2 -7% 8% 7.5%

C7H6 + H → C6H5CH2 x2, ÷10 -13% 18% 15.5% C6H5CH2 → C7H6 + H ± 21% 9% -8% 8.5% Total 2-σ R.S.S. uncertainty: ±23%

103

Appendix C: CO2 thermometry

C.1 Introduction The development of optical diagnostic techniques for the characterization of

shock-heated flows has long been a topic of interest to the chemical kinetics, hypersonics,

and supersonic combustion research communities. Characterization of the temperature

time-history behind shock waves with microsecond time-resolution is of particular

importance in understanding the effects of non-ideal gasdynamics on measurements made

in shock-heated flows. For example, high-temperature reaction rate coefficients are often

measured in shock tubes and can be quite sensitive to temporal changes in the

temperature behind the shock front. Characterization of the temperature time-history

behind the shock wave allows quantification of the uncertainties in rate coefficients

determined in shock tube experiments. Additionally, knowledge of the temperature time-

history can allow for an extension of the usable test time behind shock waves.

Early temperature measurements made behind shock waves used line reversal and

two-line emission techniques, where the concentration of the emitting species need not be

known [176]. These techniques typically employ atomic lines rather than molecular lines

due to the greater oscillator strengths of atomic species. Thus, an atomic species must be

naturally present in the shock-heated mixture or one must be seeded, a situation that may

or may not be desirable.

More recently scanned-wavelength laser absorption, pioneered by Hanson and co-

workers [177-180], has been used for temperature measurements behind shock waves.

These methods involve rapidly scanning a narrow-linewidth continuous wave laser

source over two absorption features; the resulting line-intensity ratio provides

information about the population distribution of the corresponding molecular state and

thereby enables the temperature to be inferred. Like emission techniques, the absorption

line-intensity ratio method provides temperature independent of a known concentration.

Hanson and co-workers have made temperature measurements in shock-heated flows

105

using scanned-wavelength absorption of hydroxyl radicals (OH) at 306 nm [177-178] and

nitric oxide (NO) at 225 nm [178-179]. Additionally, Phillippe and Hanson [180]

reported temperature measurements made behind shock waves using wavelength-

modulation absorption spectroscopy of molecular oxygen (O2) at 760 nm.

Previous temperature measurements made behind shock waves using laser

absorption have been limited by the tuning rate of the laser sources to a temporal

resolution of 3 to 10 kHz [177-180]. Here a fixed-frequency laser absorption technique is

presented that provides 1 MHz temporal resolution. The technique takes advantage of the

spectrally smooth absorption due to CO2 in the ultraviolet region (190-310 nm) [112], see

Chapters 2 and 5, and therefore is insensitive to collisional broadening that often limits

scanned-wavelength diagnostics at elevated pressures. In fact this diagnostic is best suited

for high-temperatures and pressures, a regime where many other techniques fail.

At room temperature CO2 is transparent at wavelengths longer than 205 nm [181],

but at high-temperature (>1000 K) CO2 has significant absorption in the 190 to 310 nm

spectral range due to vibrationally excited molecules [112,156]. Jeffries et al. [107] and

Mattison et al. [108] have investigated the feasibility of a thermometry technique using

the strongly temperature-dependent CO2 absorption cross-section in this region. Here this

technique is demonstrated for the measurement of temperature in both non-reacting and

reacting (methane ignition) shock-heated mixtures.

C.2 CO2 temperature diagnostic The ultraviolet absorption by vibrationally excited CO2 has been previously

discussed in Chapter 2. The spectrally smooth nature of the absorption and its strong

temperature-dependence enables two-wavelength absorption-based thermometry in high-

temperature gases where CO2 is present. Additionally, the absorption cross-section shows

no pressure dependence in experiments up to 8 atm and none is expected at higher

pressures, due to its smooth pre-dissociative nature, enabling high-pressure temperature

measurements.

Absorption of a narrow-linewidth laser source along a uniform path can be

described by the Beer-Lambert law

absorbance = ln(I / I0) = -σ n L

106

where I is the transmitted laser intensity, I0 is the incident beam intensity, σ

[cm2molecule-1] is the absorption cross-section, n [molecules cm-3] is the number density

of the absorbing species, and L [cm] is the absorption path length. The CO2 absorption

cross-section σCO2(λ,T) is independent of pressure, in the wavelength range of interest,

and can be described using a semi-empirical form

ln σCO2(λ, T) = a + bλ

given in Chapter 2.

Taking the ratio of absorbance at two different laser wavelengths gives

)(),(),(

),(),(

)()(

22

12

222

212

2

1 TRTT

LnTLnT

absorbanceabsorbance

CO

CO

COCO

COCO ===λσλσ

λσλσ

λλ

allowing the measurement of temperature, independent of knowledge of CO2

concentration or path length. For the measurements presented here laser radiation at 244

and 266 nm was chosen, see Figure 2.8. These two wavelengths can be conveniently

generated with available laser sources, avoid most major interference absorption, and

have an appropriate wavelength separation for measurement of temperature in the region

of interest in shock wave experiments and combustion studies. A polynomial expression

for temperature as a function of the absorbance ratio (cross-section ratio) from 1500-4500

K is given by

T = (2.857 x 104)R4 – (3.043 x 104)R3 + (1.478 x 104)R2 + (9.57 x 102)R + 1041 [K]

where R is the ratio of 266 to 244 nm absorbance (R = absorbance(266 nm) / absorbance

(244 nm)).

The CO2 absorption cross-section at 244 and 266 nm is plotted versus temperature

in Figure 2.12 and the ratio of 266 to 244 nm absorbance is plotted versus temperature,

using above equation, in Figure C.1. The uncertainty in temperature measurements made

using the absorbance ratio method is limited by the magnitude of the fractional

absorption, particularly at the more weakly absorbing wavelength 266 nm. The measured

ratio has greater uncertainty at low-temperature due to smaller absorbance (lower signal-

to-noise), and therefore the inferred temperature is more uncertain at lower temperatures.

An uncertainty in absorbance, due to signal-to-noise limitations, of ±0.1% (-ln(I/I0) =

±0.001) is generally achievable in shock wave experiments regardless of the magnitude

of the absorbance. The uncertainty in measured temperature is mapped as a function of

107

temperature and the product of partial CO2 pressure and path length (PCO2L) in Figure

C.2; notice that the uncertainty decreases at higher temperature and pressure. The

uncertainties given assume an uncertainty of ±0.1% in absorbance (-ln(I/I0) = ±0.001) and

the contours represent the percentage uncertainty in the measured temperature. The

favorable uncertainty contours in Figure C.2 demonstrate the attractiveness of this

diagnostic for high-pressure, high-temperature measurements.

C.3 Experimental The experiments described here were performed using the optical setup shown in

Figure 2.8 and the shock tube described in Chapter 2. Laser radiation at 244.061 nm (~5

GHz linewidth) and 266.075 nm (<5 MHz linewidth) was produced by the single pass of

a focused laser beam at 488 (488.122) nm (Ar+ line) and 532 (532.15) nm (Nd:YVO4)

through angle-tuned BBO as described in Chapter 2. The two UV laser beams (1.5 mW

each) were split into two components: one component, less than 1 mm in diameter,

passing through the shock tube windows (UV fused silica) at a location 2 cm from the

endwall, and one detected prior to absorption as a reference. The beams were detected

using amplified S1722-02 Hamamatsu silicon photodiodes (risetime < 1.0 µs, 4.1 mm

diameter) and recorded on a digital oscilloscope.

Experiments were performed with several different test gas mixtures to

demonstrate the diagnostic and to characterize the effect of energy release behind shock

waves. Experiments were first performed in test mixtures of 5 and 10% CO2 dilute in

argon. Finally, experiments were performed in several stoichiometric mixtures of

CH4/O2/CO2/Ar to demonstrate the diagnostic in experiments with varying levels of

energy release due to chemical reactions.

C.4 Results and discussion An example non-reacting experiment is shown in Figures C.3-C.5 for a 10%

CO2/Ar test mixture. Figure C.3 shows the measured absorbance at 244 and 266 nm and

the pressure measured with the Kistler piezoelectric pressure transducer. Figure C.4

shows the temperature inferred with the 266/244 absorbance ratio method described

108

above. The temperature is fairly constant for 2 ms before several abrupt changes due to

wave interactions. These changes are due to interaction of the reflected shock wave with

the contact surface (helium/test gas interface) and interaction of the reflected shock wave

with the rarefaction wave (expansion of helium driver gas). Additionally, a comparison of

the measured temperature is made with the temperature calculated from the pressure trace

via the assumption of an isentropic relationship between pressure and temperature

γγ 1

00

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

PPTT

A comparison of the temperature measured with the absorbance ratio method and that

resulting from the isentropic assumption shows excellent agreement (Figure C.4), with a

difference in the two of no greater that 2% for this experiment (Figure C.5). Note that the

specific heat ratio, γ, for the 10% CO2/Ar mixture (Figure C.4) varies only 0.02% over

the temperature range (1800-2200 K) of this experiment. Therefore, the isentropic

relationship given above, assuming a constant γ is valid. This agreement serves both to

validate the CO2 temperature sensor and to confirm that for shock wave experiments with

no energy release the isentropic assumption holds; other studies have shown similar

results [177,186]. Thus in typical reaction rate experiments where there is negligible

energy release, changes in post-shock temperature can be inferred directly from the

pressure trace, without optical measurements. This result emphasizes the potential utility

of precise, quantitative pressure measurements during reaction rate experiments where

small uncertainties in temperature can lead to large errors in inferred rates. For example,

a 1% uncertainty in temperature at 1500 K (±15 K) can lead to a 30% uncertainty in the

measured dissociation rate coefficient for a molecule with an activation energy of 80

kcal/mol.

A comparison between the temperature measured immediately after the passage

of the reflected shock and that calculated with the ideal normal shock relations is given

for the entire set of experiments in Figure C.6. The scatter in the difference between the

measured temperature and the ideal temperature has a 1-σ standard deviation of 0.6% and

is no greater than 2.3%. Additionally, the scatter shows no trend over a wide range of

fractional absorption, demonstrating the precision of the thermometry measurement

technique for a range of conditions. Note, the CO2 absorbance at 266 and 244 nm ranges

109

from 0.5 to 13% and 2 to 32%, respectively, illustrating the large dynamic range of

conditions investigated. Because the CO2 absorption cross-section was measured behind

shock waves in the same experimental facility, the excellent agreement between the

measured temperature and calculated temperature is not a complete confirmation of the

accuracy of the technique. Rather, the agreement is an indication of the consistency of the

laser absorption technique and shock velocity measurement, which gives the ideal shock

temperature. However, because the shock wave calculation for temperature has an

uncertainty of <1% the agreement between measured temperature and calculated

temperature demonstrates that the measurement scatter is within the uncertainties given

in Figure C.1.

To demonstrate the CO2 absorption technique for thermometry in combustion

systems, and to characterize the effects of energy release on temperature in shock tube

studies, a set of ignition experiments in CH4/O2/CO2/Ar mixtures were perfomed. Three

mixtures were used: (1) 1% CH4/ 2% O2/ 10% CO2/ dilute in argon, (2) 2% CH4/ 4% O2/

10% CO2/ dilute in argon, and (3) 4% CH4/ 8% O2/ 10% CO2/ dilute in argon.

Prior to the ignition experiments the absorption cross-sections of O2 and CH4 at

the two wavelengths of interest, 244.061 and 266.075 nm, were measured. Absorption by

CH4 at these wavelengths was undetectable but absorption by hot O2 was detectable and

the measured cross-sections are given in Figure C.7 for ~1 atm. In the shock wave

ignition experiments presented here the interference at 266.075 nm is negligible and

needs not be accounted for, but the larger interference at 244.061 nm was accounted for.

The absorption by O2 at 244.061 nm was estimated (with a chemical model described

below) and subtracted from the overall measured 244.061 nm absorbance to obtain the

portion of the signal due to CO2 absorption. This correction causes greater uncertainty in

the resulting temperature measurements relative to the non-reacting case. These

corrections may or may not be needed in practical combustion thermometry depending on

the relative CO2 to O2 concentrations. Also, the interference due to O2 can be avoided

with a continuous-wave laser source which can be tuned in wavelength to an O2

absorption minimum. Simulations of the O2 spectrum at 2000 K and 1 atm in this

wavelength range have been carried out using the LIFSIM calculator developed by

Bessler et al. [187]. The results for the O2 spectrum around 244 nm are given in Figure

110

C.8. The current laser source sits in the wing of an absorption feature, but the interference

could be reduced by a factor of seven by tuning between two absorption lines to 243.99

nm.

In Figure C.9 the absorbance at 266 and 244 nm (measured and corrected for O2

interference) is given for an example experiment; the O2 contribution to the absorption

ranges from 4 to 15% of the measured total absorption. For the three different test

mixtures the resulting temperature time-histories are given in Figure C.10 with

comparison to model results calculated using a constant volume constraint and the GRI

Mech 3.0 methane combustion mechanism [95]. The correction applied to 244 nm

absorbance for the O2 interference was also determined by using the output of the GRI

Mech 3.0 mechanism for the O2 mole fraction and the measured temperature dependence

of the O2 absorption cross-section.

A comparison between constant volume model calculations and measured

temperature (Figure C.10) shows good agreement for the ignition delay and the overall

trends in temperature. However, the post-ignition temperature plateau measured with

laser absorption is lower, in all three cases, than that predicted with the constant-volume

model. This shortcoming of the constant volume model is expected because as energy is

released due to ignition and the test gas temperature begins to rise at the test section

location (2 cm from endwall) the hot combustion gases will expand down the shock tube

resulting in a somewhat lower temperature plateau. (Note that the measured lower

temperature in the post-ignition plateau was accounted for when determining the O2

interference absorption cross-section at 244 nm.) For the examples given in Figure C.10

the measured post-ignition temperature rise is 60 to 75% of that predicted by the constant

volume model. Additionally, in the bottom graph of Figure C.10 the steep rise in

temperature at ignition is due to the passage of a blast wave emanating from the shock

tube endwall. When shock-heated fuel/oxidizer mixtures are of great enough energy

content a blast wave is formed at the endwall and propagates down the shock tube

reaching the optical location before the mixture is able to auto-ignite [188]. This is a

common problem with ignition time measurements made at sidewall locations in

energetic mixtures. Davidson and Hanson [189] have recently reported measurements of

the post-ignition pressure plateaus in shock wave ignition experiments for iso-octane and

111

n-heptane. Consistent with their findings, we found temperature plateaus lower than that

predicted with the constant-volume model. The current temperature measurements

provide needed targets for improved models for reflected shock waves in reacting gases.

A model that includes both the finite-rate chemistry and transport in the shock tube is

needed to accurately model experiments with significant energy release, such as these.

C.5 Summary A thermometry technique with microsecond time-resolution is demonstrated in

shock wave experiments. The technique uses the highly temperature-dependent

spectrally-smooth ultraviolet absorption feature of CO2. The ratio of absorbance

measured at 266 and 244 nm with fixed frequency continuous-wave laser sources is used

to infer the temperature. The technique is demonstrated by monitoring temperature in

non-reacting CO2/argon mixtures after shock-heating. A comparison between the

measured temperature and that resulting from an isentropic assumption between pressure

(measured with a piezoelectric transducer) and temperature shows excellent agreement.

Temperature measurements were also made in shock wave experiments with energy

release due to the ignition of CH4/O2/CO2/argon mixtures. It was found that the measured

post-ignition temperature plateau was somewhat lower than that predicted with a

constant-volume model due to expansion of the hot combusted gases down the shock

tube.

112

1500 2000 2500 3000 3500 4000 45000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

R =

σ26

6 / σ 24

4

Temperature [K]

Figure C.1: Ratio of 266 to 244 nm absorption cross-section; also referred to as absorbance ratio, R.

1500 2000 2500 30000.1

1

10

100

0.5 %

1 %2 %

5 %

PC

O2 L

[atm

cm

]

Temperature [K]

10 %

Figure C.2: Uncertainty in measured temperature for given temperature and product of partial CO2 pressure and path length (PCO2L). The contours represent lines of constant uncertainty in the temperature measurement assuming an uncertainty in measured absorbance of ±0.1% (-ln(I / I0) = ±0.001).

113

0 1000 2000 3000 4000 5000

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Pre

ssur

e [a

tm]

Abs

orba

nce

[ -ln

(I / I

0) ]

Time [µs]

Pressure

244 absorbance

266 absorbance

Figure C.3: Example non-reacting experiment: measured absorbance at 266 and 244 nm and measured pressure. Initial reflected shock conditions from ideal shock relations: 10% CO2/Ar, 2138 K and 1.226 atm.

0 1000 2000 3000 4000 50001800

1900

2000

2100

2200

2300

Tem

pera

ture

[K]

Time [µs]

Ideal ShockRelations

Isentropic- frompressure (light line)

Absorptionmeasurement(heavy line)

Figure C.4: Measured temperature (dark line) vs isentropic temperature (light line) for experiment in Figure C.3. Measured temperature determined from absorbance ratio (absorbance traces given in Figure C.3). Isentropic temperature determined from pressure measurement (given in Figure C.3).

114

0 1000 2000 3000 4000 5000-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

(Tm

easu

red -

T isen

tropi

c) / T

isen

tropi

c

Time [µs]

Figure C.5: Comparison of measured temperature to isentropic temperature from the results given in Figure C.4.

1600 1800 2000 2200 2400 2600 2800 3000-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

5% CO2 / Ar 10% CO2 / Ar

(Tm

easu

red -

Tca

lcul

ated

) / T

calc

ulat

ed

Temperature [K]

Figure C.6: Comparison of measured post-shock temperature to temperature calculated with the ideal shock relations. Post-shock vibrational equilibrium was assumed.

115

1600 1800 2000 2200 2400 2600 280010-23

10-22

10-21

10-20

10-19

266.061 nm

σO

2 [cm

2 mol

ecul

e-1]

Temperature [K]

244.075 nm

Figure C.7: Molecular oxygen absorption cross-section at 244.061 and 266.075 nm (1 atm).

243.90 243.95 244.00 244.05 244.1010-22

10-21

10-20

10-19

σ O2 [c

m2 m

olec

ule-1

]

Wavelength [nm]

Figure C.8: Molecular oxygen absorption spectrum around 244 nm for 2000K and 1 atm, calculated using LIFSIM calculator [187]. Current doubled Ar+ line is at 244.061 nm, a source at 243.99 nm would reduce the interference due to O2 by a factor of seven.

116

-100 0 100 200 300 400 500

0.00

0.05

0.10

0.15

0.20

Abs

orba

nce

[-ln(

I/I0)]

Time [µs]

266 nm

244 nm

244 nm corrected

Figure C.9: Example absorbance for energetic ignition experiment. Initial reflected shock conditions: 1% CH4/ 2% O2/ 10% CO2/ Ar, 2085 K and 1.297 atm. 266 nm and 244 nm measured absorbance, dark lines; corrected (O2 interference) 244 nm absorbance, light line.

117

0 100 200 300 400 500

1600

2000

2400

2800

1600

2000

2400

2800

0 100 200 300 400 5001600

2000

2400

2800

Time [µs]

Tem

pera

ture

[K]

Time [µs]

Figure C.10: Measured temperature (solid thick lines) versus constant volume calculation (solid thin lines). Top graph (initial reflected shock conditions): 1% CH4/ 2% O2/ 10% CO2/ Ar, 2085 K, and 1.297 atm. Middle graph: 2% CH4/ 4% O2/ 10% CO2/ Ar, 1890 K, 1.358 atm. Bottom graph: 4% CH4/ 8% O2/ 10% CO2/ Ar, 1851 K, 1.277 atm. Post-shock vibrational equilibrium was assumed. Error bars represent a 1-σ confidence interval.

118

Appendix D: Azomethane synthesis

This appendix describes the process used to synthesize azomethane. The method

used is that of Jahn [99] and the below instructions were developed by David F. Davidson

at Stanford University.

D.1 Chemicals The following chemicals are needed:

1) Sodium acetate (MW 82.03 Aldrich# 24,124-5)

2) Copper chloride (II) dihydrate (FW 170.48 Aldrich# 22,178-3)

3) 1,2-dimethylhydrazine dihydrochloride (DMHZ, MW 133.02 Aldrich#

D16,180-2)

4) Distilled water

D.2 Supplies The following supplies are needed to make and hold solutions:

1) 1 liter graduated cylinder

2) funnel

3) Kimwipes

4) scale/balance

5) weighing papers

6) bottle to hold sodium acetate solution 1-2 liter with wide mouth cap

7) bottle to hold copper chloride solution 1-2 liter with wide mouth cap

8) plastic bottle to hold waste solution 1 gallon with cap

9) secondary containment basin for chemicals and bottles

The following supplies are needed to make brown mud:

1) 1 liter beaker, open mouth

2) glass stirring rod/ rubber policeman

119

The following supplies are needed to dry brown mud:

1) long spoon

2) Buchner funnel, and seal

3) Florence flask with vacuum connection

4) filter paper for Buchner funnel

5) vacuum hose connection and vacuum pump

The following supplies are needed to heat brown mud:

1) flat bottom heating flask with connection to vacuum manifold

2) oil dish for heating flask

3) heating plate

4) thermometer to 150º C

The following supplies are needed to trap azomethane:

1) long neck AirFree flask and valve

2) LN2 dewar for flask

3) larger dewar for LN2 supply

4) glass manifold for flask, pump and heating flask clamps and orings vacuum

line adapters for 3 connections

5) two chemistry stands and clamps

D.3 Directions for azomethane production 1) Mix 1 molar NaCH3CO2 solution (MW 82.03) in one liter of distilled water.

2) Mix 1 molar CuCl2:2H2O solution (FW 170.48) in one liter of distilled water.

3) Dissolve fully 10 grams DMHZ in 300 cc of sodium acetate solution.

4) Add 300 cc of cupric chloride solution.

5) Brown mud azomethane:(CuCl)2 complex will precipitate.

6) Wash through aspirated Buchner funnel/paper filter with 1 liter of water.

7) Dry at room temperature by aspirating through funnel or free-standing.

8) Break into two amounts. Store one for later use.

9) Put in heating flask, break up and press onto bottom of flask to form even

layer.

10) Then heat in oil bath to 125-150 º C to release azomethane trapping with LN2.

120

11) Heating flask cleans easily with water.

D.4 Azomethane yield The 10 grams (0.0752 moles) of DMHZ is fully dissolved in the aqueous sodium

acetate solution and then oxidized by the cupric chloride solution resulting in a

precipitate, (CuCl)2:(CH3)NN(CH3). The precipitate then is heated causing its thermal

decomposition

(CuCl)2:(CH3)NN(CH3) → (CuCl)2 + (CH3)NN(CH3)

Assuming a 70% yield, per Jahn [99], the 0.0752 of DMHZ gives 0.0526 moles of

azomethane (3.0 grams) which at STP is 1.2 liters or in a 100 mL flask is 12 atm. This

method provides very clean gaseous azomethane for making mixtures because any other

impurities (CH3CO2, (CuCl)2:(CH3)NN(CH3), that make their way to the LN2 trap will

have very low vapor pressures in comparison to the azomethane and will not vaporize

when making mixtures.

121

Appendix E: Mechanism used for benzyl

and toluene decomposition

Table E.1: Reaction mechanism used for modeling toluene and benzyl decomposition

experiments, rate coefficients in the form k = ATb exp(-Ea / RT).

Reaction A

[cm, mol, s] b

Ea

[cal/mol] Ref.

(7) C6H5CH2 → C7H6 + H (1.5 atm) 8.20 x 10-14 0 80670 this study

(8) C6H5CH3 → C6H5CH2 + H (1.5 atm) 2.09 x 1015 0 87507 this study

(9) C6H5CH3 → C6H5 + CH3 (1.5 atm) 2.66 x 1016 0 97880 this study

(12) C6H5CH2I → C6H5CH2 + I 5.89 x 1014 0 43260 [82]

(14) C6H5CH2 +H → C6H5CH3 2.00 x 1014 0 0 [190]

(15) C7H6 + H → C6H5CH2 1.00 x 1014 0 0 [74]

(16) C6H5CH3 +H → C6H5CH2 + H2 1.26 x 1015 0 14818 [81]

C6H5 + H → C6H6 7.80 x 1013 0 0 [191]

H + C6H5CH3 → CH3 + C6H6 5.78 x 1013 0 8090 [190]

CH3 + C6H5CH3 → C6H5CH2 + CH4 2.51 x 1010 0 6940 [192]

C6H5 + C6H5CH3 → C6H6 + C6H5CH2 7.94 x 1013 0 11940 [193]

(13) C6H5CH2 + C6H5CH2 → C6H5CH2CH2C6H5 2.51 x 1012 0.4 0 [82]

C6H5CH2CH2C6H5 → C6H5CH2 + C6H5CH2 7.94 x 1014 0 59751 [82]

C6H5CH2CH2C6H5 → C6H5CH2CHC6H5 + H 1.00 x 1016 0 83660 [194]

C6H5CH2CH2C6H5 + H → C6H5CH2CHC6H5

+ H23.16 x 1012 0 0 [91]

C6H5CH2CHC6H5 → C6H5CHCHC6H5 + H 7.94 x 1015 0 51864 [195]

C6H5CH2 + I → C6H5CH2I 5.01 x 1013 0 0 [82]

I + C6H5CH2I → C6H5CH2 + I2 5.62 x 1011 0 3500 [196]

C6H5CH2 + I2 → I + C6H5CH2I 5.62 x 1010 0 0 [196]

I2 → 2I 9.79 x 1013 0 30400 [197]

I + I → I2 7.01 x 1015 -0.45 420 [198]

123

(1) C2H6 ↔ 2CH3, high-pressure-limit 1.88 x 1050 -9.72 107338 This study

low-pressure-limit 3.72 x 1065 -13.14 101575

Troe parameters a = 0.39, b = 100.0, c = 1900.0, d = 6000.0

H + C2H5 ↔ C2H6, high-pressure-limit 3.65 x 1017 -0.99 1580 [95]



H + C2H6 ↔ C2H5 + H2 1.15 x 108 1.90 7530 [95]

H + CH3 ↔ CH4, high-pressure-limit 9.73 x 1015 -0.534 536 [95]



H + CH4 ↔ CH3 + H2 6.60 x 108 1.62 10840 [95]

H + C2H4 ↔ C2H5, high-pressure-limit 3.78 x 1011 0.454 1820 [95]



CH3 + C2H6 ↔ C2H5 + CH4 6.14 x 106 1.74 10450 [95]

124

Appendix F: Shock tube gasdynamics

The gasdynamic wave interactions that occur during shock tube experiments can

be described using x-t diagrams. Assuming ideal inviscid compressible flow, the shock

tube can be modeled using the simple one-dimensional wave propagation equations

derived from conservation of mass, momentum, and energy [176,203]. The resulting

reflected shock test-time is limited at high-temperatures (strong shocks) by the interaction

of the reflected shock wave with the driver-driven gas contact surface. At low-

temperatures (weak shocks) the test-time is limited by the interaction of the reflected

shock wave with the rarefaction (expansion) head that is generated by the breakage of the

diaphragm and the subsequent blow down of the high-pressure driver gas.

F.1 High-temperature test-time In the high-temperature (strong shock) case, when the test-time is limited by the

reflected shock wave interaction with the contact surface, three possible wave

interactions can occur. When the speed of sound in region 2 (incident condition) is

greater than the speed of sound in region 3 (condition behind the contact surface), a2 > a3,

the reflected shock wave decelerates when it passes through the contact surface

generating a second shock wave that travels back towards the endwall and reflects off of

it causing the pressure at the test-location to incrementally increase. See Figure F.1 for a

x-t schematic and the top graph in Figure F.5 for an example pressure trace for this case.

When the speed of sound in region 2 is greater than the speed of sound in region

3, a3 > a2, the reflected shock wave accelerates when it passes through the contact surface

generating a rarefaction wave that propagates back to the endwall and reflects off of it.

This rarefaction wave then generates another shock wave when it interacts with the

contact surface. These wave interactions cause the pressure at the test-location to initially

decrease, due to the rarefaction wave generated by the reflected shock interaction with

the contact surface. However, at later times the pressure will increase due to the

125

generation of a shock wave when this newly created rarefaction wave interacts with the

contact surface. These types of wave interactions are responsible for the temperature and

pressure profiles shown in Figures C.3 and C.4 where the pressure falls and rises after

about 2 ms of test-time. See Figure F.2 for a x-t schematic and the middle graph in Figure

F.5 for an example pressure trace for this case. Note that the final decrease in pressure at

3 ms (Figure F.5) is due to the rarefaction head propagating through the test-location.

Finally, if the sound speeds in region 2 and 3 are equal, a2 = a3, the reflected

shock wave passes through the contact surface without a change in speed and the contact

surface is brought to rest allowing significantly longer test times. This limiting case is

called shock tailoring; see Figure F.3 for a x-t schematic of this case.

F.2 Low-temperature test-time When the diaphragm breaks and the incident shock wave is generated, a

rarefaction head is launched towards the driver endwall. This wave reflects off the driver

endwall and proceeds down the shock tube following the incident shock and contact

surface. For cases when the incident shock speed is slow, the rarefaction head can pass

through the contact surface and directly interact with the reflected shock wave. In this

case the pressure at the test-location simply decays when the rarefaction wave reaches it.

This happens for cases when the reflected shock temperature is low (weak shock). See

Figure F.4 for a x-t schematic and the bottom graph in Figure F.5 for an example pressure

trace for this case.

F.3 Reflected shock test-time and pressure traces Experimental pressure traces for three of the cases described above are shown in

Figure F.5. The three pressure traces shown in the figure illustrate that the pressure

slightly increases during the test-time until reflected shock interactions with either the

contact surface or the rarefaction wave disrupt the test-time causing a more drastic

discontinuous decrease or increase in the pressure. The slight increase in the pressure

during the test-time is due to non-ideal effects behind the reflected shock wave. After the

shock wave reflects off the endwall, it travels into a gas with non-uniformities due to

126

incident shock wave attenuation. These non-uniformities cause small compression waves

to propagate back to the test-location and thus cause a slight increase in pressure and

temperature during the test-time; see Petersen Ph.D. thesis [175] for details.

The three pressure traces in Figure F.5 show the limitation of the test-time by the

three different wave interactions shown in the x-t diagrams given in Figures F.1, F.2, and

F.4. The test-time was determined from the experimental pressure traces by estimating

the time at which the discontinuous change in pressure occurs. The experimental test-

time is plotted versus reflected shock temperature for several experiments in Figure F.6.

Note that the test-time peaks around 1800 K and falls off to low- and high-temperature

due to reflected shock interaction with the rarefaction head and contact surface

respectively.

127

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

Figure F.1: Schematic of x-t diagram for high-temperature (strong shock) case where a2 > a3.

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

Figure F.2: Schematic of x-t diagram for high-temperature (strong shock) case where a3 > a3.

128

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

Figure F.3: Schematic x-t diagram for high-temperature (strong shock) case where a2 = a3, tailored condition.

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

t

xdiaphragm

incident shock

rarefaction wave

contact surface

test-time

reflected shock

Figure F.4: Schematic x-t diagram for low-temperature (weak shock) case where reflected shock interaction with rarefaction wave limits test-time.

129

0 1000 2000 3000 4000

0.00.20.40.60.81.01.21.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1000 2000 3000 40000.0

0.2

0.4

0.6

0.8

1.0

1.2

test-time

test-time

test-time

Time [µs]

Reflected shock-contact surface interactiona2 > a3

Reflected shock-rarefaction wave interaction

Reflected shock-contact surface interactiona3 > a2

Rel

ativ

e P

ress

ure

Time [µs]

Figure F.5: Example pressure traces measured with KISTLER piezoelectric transducer. Top graph: example high-temperature (strong shock) case where a2 > a3 (Figure F.1 x-t diagram). Middle graph: example high-temperature (strong shock) case where a3 > a2 (Figure F.2 x-t diagram). Bottom graph: example low-temperature case (Figure F.4 x-t diagram).

130

1000 1500 2000 2500 30000.0

0.5

1.0

1.5

2.0

2.5

3.0

rarefraction wavelimited test-time

contact surfacelimited test-time

Test

-tim

e [m

s]

T5 [K]

Figure F.6: Measured test-time as a function of reflected shock temperature for helium

driven argon shocks.

131

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